Frenzy007:
I would like it if u can help ne solve it by snapping ur solution with ur phone or typing it here cos i know the answer but i need to know how to solve it.






X+3/X-2 - 1-X/x =17/4

that is x+3 over x-2 then minus 1-x over x = 17 over 4
thank you.

Frenzy007:
I would like it if u can help ne solve it by snapping ur solution with ur phone or typing it here cos i know the answer but i need to know how to solve it.






X+3/X-2 - 1-X/x =17/4

that is x+3 over x-2 then minus 1-x over x = 17 over 4
thank you.

Frenzy007:
I would like it if u can help ne solve it by snapping ur solution with ur phone or typing it here cos i know the answer but i need to know how to solve it.






X+3/X-2 - 1-X/x =17/4

that is x+3 over x-2 then minus 1-x over x = 17 over 4
thank you.

Frenzy007:
cc
MIKEZURUKI
NEXTPRINCE
tempest01
Biafraisdead
raintaker
tanx

Frenzy007:
cc
MIKEZURUKI
NEXTPRINCE
tempest01
Biafraisdead
raintaker
tanx

biafraisdead:

x=4 or x=-2/9. it's a simple quadratic equation.

Deicide:
The answers are in surd/decimal forms sha the last eqn after simplifying is: -9x2 + 10x + 8 = 0

Raintaker:
too long
..




 x+3/x-2-1-x/x=17/4 

Two solutions were found :

 x =(33-√897)/8= 0.381 x =(33+√897)/8= 7.869

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 

                     x+3/x-2-1-x/x-(17/4)=0 

Step by step solution :

Step  1  :

17 Simplify —— 4

Equation at the end of step  1  :

3 x 17 ((((x+—)-2)-1)-—)-—— = 0 x x 4

Step  2  :

x Simplify — x

Equation at the end of step  2  :

3 17 ((((x+—)-2)-1)-1)-—— = 0 x 4

Step  3  :

3 Simplify — x

Equation at the end of step  3  :

3 17 ((((x + —) - 2) - 1) - 1) - —— = 0 x 4

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  x  as the denominator :

x x • x x = — = ————— 1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • x + 3 x2 + 3 ————————— = —————— x x

Equation at the end of step  4  :

(x2 + 3) 17 (((———————— - 2) - 1) - 1) - —— = 0 x 4

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

2 2 • x 2 = — = ————— 1 x

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = x2 + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q  then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  3. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      4.00        -3     1      -3.00      12.00        1     1      1.00      4.00        3     1      3.00      12.00   


Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions 

(x2+3) - (2 • x) x2 - 2x + 3 ———————————————— = ——————————— x x

Equation at the end of step  5  :

(x2 - 2x + 3) 17 ((————————————— - 1) - 1) - —— = 0 x 4

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

1 1 • x 1 = — = ————— 1 x

Trying to factor by splitting the middle term

 6.2     Factoring  x2 - 2x + 3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -2 .

     -3   +   -1   =   -4     -1   +   -3   =   -4     1   +   3   =   4     3   +   1   =   4


Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 6.3       Adding up the two equivalent fractions 

(x2-2x+3) - (x) x2 - 3x + 3 ——————————————— = ——————————— x x

Equation at the end of step  6  :

(x2 - 3x + 3) 17 (————————————— - 1) - —— = 0 x 4

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

1 1 • x 1 = — = ————— 1 x

Trying to factor by splitting the middle term

 7.2     Factoring  x2 - 3x + 3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -3 .

     -3   +   -1   =   -4     -1   +   -3   =   -4     1   +   3   =   4     3   +   1   =   4


Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions 

(x2-3x+3) - (x) x2 - 4x + 3 ——————————————— = ——————————— x x

Equation at the end of step  7  :

(x2 - 4x + 3) 17 ————————————— - —— = 0 x 4

Step  8  :

Trying to factor by splitting the middle term

 8.1     Factoring  x2-4x+3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -4x  its coefficient is  -4 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -4 .

     -3   +   -1   =   -4   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -3  and  -1 
                     x2 - 3x - 1x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-3)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x-3)
Step-5 : Add up the four terms of step 4 :
                    (x-1)  •  (x-3)
             Which is the desired factorization

Calculating the Least Common Multiple :

 8.2    Find the Least Common Multiple 

      The left denominator is :       x 

      The right denominator is :       4 

        Number of times each prime factor
        appears in the factorization of: Prime 
 Factor  Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right} 2022 Product of all 
 Prime Factors 144

                  Number of times each Algebraic Factor
            appears in the factorization of:    Algebraic    
    Factor     Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right}  x 101


      Least Common Multiple: 
      4x 

Calculating Multipliers :

 8.3    Calculate multipliers for the two fractions 


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 4

   Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

 8.4      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent, y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well. 

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. (x-1) • (x-3) • 4 —————————————————— = ————————————————— L.C.M 4x R. Mult. • R. Num. 17 • x —————————————————— = —————— L.C.M 4x

Adding fractions that have a common denominator :

 8.5       Adding up the two equivalent fractions 

(x-1) • (x-3) • 4 - (17 • x) 4x2 - 33x + 12 ———————————————————————————— = —————————————— 4x 4x

Trying to factor by splitting the middle term

 8.6     Factoring  4x2 - 33x + 12 

The first term is,  4x2  its coefficient is  4 .
The middle term is,  -33x  its coefficient is  -33 .
The last term, "the constant", is  +12 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 12 = 48 

Step-2 : Find two factors of  48  whose sum equals the coefficient of the middle term, which is   -33 .

     -48   +   -1   =   -49     -24   +   -2   =   -26     -16   +   -3   =   -19     -12   +   -4   =   -16     -8   +   -6   =   -14     -6   +   -8   =   -14


For tidiness, printing of 14 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Equation at the end of step  8  :

4x2 - 33x + 12 —————————————— = 0 4x

Step  9  :

When a fraction equals zero :

 9.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

4x2-33x+12 —————————— • 4x = 0 • 4x 4x

Now, on the left hand side, the  4x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   4x2-33x+12  = 0

Parabola, Finding the Vertex :

 9.2      Find the Vertex of   y = 4x2-33x+12

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 4 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   4.1250  

 Plugging into the parabola formula   4.1250  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 4.13 * 4.13 - 33.0 * 4.13 + 12.0 
or   y = -56.063

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x2-33x+12
Axis of Symmetry (dashed)  {x}={ 4.13} 
Vertex at  {x,y} = { 4.13,-56.06}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.38, 0.00} 
Root 2 at  {x,y} = { 7.87, 0.00} 

Solve Quadratic Equation by Completing The Square

 9.3     Solving   4x2-33x+12 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x2-(33/4)x+3 = 0

Subtract  3  from both side of the equation :
   x2-(33/4)x = -3

Now the clever bit: Take the coefficient of  x , which is  33/4 , divide by two, giving  33/8 , and finally square it giving  1089/64 

Add  1089/64  to both sides of the equation :
  On the right hand side we have :
   -3  +  1089/64    or,  (-3/1)+(1089/64) 
  The common denominator of the two fractions is  64   Adding  (-192/64)+(1089/64)  gives  897/64 
  So adding to both sides we finally get :
   x2-(33/4)x+(1089/64) = 897/64

Adding  1089/64  has completed the left hand side into a perfect square :
   x2-(33/4)x+(1089/64)  =
   (x-(33/) • (x-(33/)  =
  (x-(33/)2 
Things which are equal to the same thing are also equal to one another. Since
   x2-(33/4)x+(1089/64) = 897/64 and
   x2-(33/4)x+(1089/64) = (x-(33/)2 
then, according to the law of transitivity,
   (x-(33/)2 = 897/64

We'll refer to this Equation as  Eq. #9.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(33/)2   is
   (x-(33/)2/2 =
  (x-(33/)1 =
   x-(33/

Now, applying the Square Root Principle to  Eq. #9.3.1  we get:
   x-(33/ = √ 897/64 

Add  33/8  to both sides to obtain:
   x = 33/8 + √ 897/64 

Since a square root has two values, one positive and the other negative
   x2 - (33/4)x + 3 = 0
   has two solutions:
  x = 33/8 + √ 897/64 
   or
  x = 33/8 - √ 897/64 

Note that  √ 897/64 can be written as
  √ 897  / √ 64   which is √ 897  / 8 

Solve Quadratic Equation using the Quadratic Formula

 9.4     Solving    4x2-33x+12 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A 

  In our case,  A   =     4
                      B   =   -33
                      C   =   12 

Accordingly,  B2  -  4AC   =
                     1089 - 192 =
                     897

Applying the quadratic formula :

               33 ± √ 897 
   x  =    ——————
                      8

  √ 897   , rounded to 4 decimal digits, is  29.9500
 So now we are looking at:
           x  =  ( 33 ±  29.950 ) / 8

Two real solutions:

 x =(33+√897)/8= 7.869 

or:

 x =(33-√897)/8= 0.381 

Supplement : Solving Quadratic Equation Directly

Solving  x2-4x+3  = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 10.1      Find the Vertex of   y = x2-4x+3

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 1 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   2.0000  

 Plugging into the parabola formula   2.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 2.00 * 2.00 - 4.0 * 2.00 + 3.0 
or   y = -1.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-4x+3
Axis of Symmetry (dashed)  {x}={ 2.00} 
Vertex at  {x,y} = { 2.00,-1.00}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 1.00, 0.00} 
Root 2 at  {x,y} = { 3.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 10.2     Solving   x2-4x+3 = 0 by Completing The Square .

 Subtract  3  from both side of the equation :
   x2-4x = -3

Now the clever bit: Take the coefficient of  x , which is  4 , divide by two, giving  2 , and finally square it giving  4 

Add  4  to both sides of the equation :
  On the right hand side we have :
   -3  +  4    or,  (-3/1)+(4/1) 
  The common denominator of the two fractions is  1   Adding  (-3/1)+(4/1)  gives  1/1 
  So adding to both sides we finally get :
   x2-4x+4 = 1

Adding  4  has completed the left hand side into a perfect square :
   x2-4x+4  =
   (x-2) • (x-2)  =
  (x-2)2 
Things which are equal to the same thing are also equal to one another. Since
   x2-4x+4 = 1 and
   x2-4x+4 = (x-2)2 
then, according to the law of transitivity,
   (x-2)2 = 1

We'll refer to this Equation as  Eq. #10.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-2)2   is
   (x-2)2/2 =
  (x-2)1 =
   x-2

Now, applying the Square Root Principle to  Eq. #10.2.1  we get:
   x-2 = √ 1 

Add  2  to both sides to obtain:
   x = 2 + √ 1 

Since a square root has two values, one positive and the other negative
   x2 - 4x + 3 = 0
   has two solutions:
  x = 2 + √ 1 
   or
  x = 2 - √ 1 

Solve Quadratic Equation using the Quadratic Formula

 10.3     Solving    x2-4x+3 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A 

  In our case,  A   =     1
                      B   =    -4
                      C   =   3 

Accordingly,  B2  -  4AC   =
                     16 - 12 =
                     4

Applying the quadratic formula :

               4 ± √ 4 
   x  =    ————
                   2

Can  √ 4 be simplified ?

Yes!   The prime factorization of  4   is
   2•2  
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 4   =  √ 2•2   =
                ±  2 • √ 1   =
                ±  2 

So now we are looking at:
           x  =  ( 4 ± 2) / 2

Two real solutions:

x =(4+√4)/2=2+= 3.000 

or:

x =(4-√4)/2=2-= 1.000 

Two solutions were found :

 x =(33-√897)/8= 0.381 x =(33+√897)/8= 7.869


Processing ends successfully

biafraisdead:

not true, after simplifying u would have: 9x2 - 34x - 8= 0. Let me give u a friendly tips; firstly assuming ur simplification above was correct don't u think it would make more sense to write it as this: 9x2 - 10x - 8 =0 rather than as: -9x2 + 10x + 8 = 0. secondly always verify ur answers by substituting ur aswers into the original equation and see if it makes sense.

Raintaker:
too long
..




 x+3/x-2-1-x/x=17/4 

Two solutions were found :

 x =(33-√897)/8= 0.381 x =(33+√897)/8= 7.869

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 

                     x+3/x-2-1-x/x-(17/4)=0 

Step by step solution :

Step  1  :

17 Simplify —— 4

Equation at the end of step  1  :

3 x 17 ((((x+—)-2)-1)-—)-—— = 0 x x 4

Step  2  :

x Simplify — x

Equation at the end of step  2  :

3 17 ((((x+—)-2)-1)-1)-—— = 0 x 4

Step  3  :

3 Simplify — x

Equation at the end of step  3  :

3 17 ((((x + —) - 2) - 1) - 1) - —— = 0 x 4

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  x  as the denominator :

x x • x x = — = ————— 1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 4.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • x + 3 x2 + 3 ————————— = —————— x x

Equation at the end of step  4  :

(x2 + 3) 17 (((———————— - 2) - 1) - 1) - —— = 0 x 4

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

2 2 • x 2 = — = ————— 1 x

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = x2 + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q  then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  3. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      4.00        -3     1      -3.00      12.00        1     1      1.00      4.00        3     1      3.00      12.00   


Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions 

(x2+3) - (2 • x) x2 - 2x + 3 ———————————————— = ——————————— x x

Equation at the end of step  5  :

(x2 - 2x + 3) 17 ((————————————— - 1) - 1) - —— = 0 x 4

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

1 1 • x 1 = — = ————— 1 x

Trying to factor by splitting the middle term

 6.2     Factoring  x2 - 2x + 3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -2 .

     -3   +   -1   =   -4     -1   +   -3   =   -4     1   +   3   =   4     3   +   1   =   4


Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 6.3       Adding up the two equivalent fractions 

(x2-2x+3) - (x) x2 - 3x + 3 ——————————————— = ——————————— x x

Equation at the end of step  6  :

(x2 - 3x + 3) 17 (————————————— - 1) - —— = 0 x 4

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Subtracting a whole from a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

1 1 • x 1 = — = ————— 1 x

Trying to factor by splitting the middle term

 7.2     Factoring  x2 - 3x + 3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -3 .

     -3   +   -1   =   -4     -1   +   -3   =   -4     1   +   3   =   4     3   +   1   =   4


Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions 

(x2-3x+3) - (x) x2 - 4x + 3 ——————————————— = ——————————— x x

Equation at the end of step  7  :

(x2 - 4x + 3) 17 ————————————— - —— = 0 x 4

Step  8  :

Trying to factor by splitting the middle term

 8.1     Factoring  x2-4x+3 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -4x  its coefficient is  -4 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -4 .

     -3   +   -1   =   -4   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -3  and  -1 
                     x2 - 3x - 1x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-3)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x-3)
Step-5 : Add up the four terms of step 4 :
                    (x-1)  •  (x-3)
             Which is the desired factorization

Calculating the Least Common Multiple :

 8.2    Find the Least Common Multiple 

      The left denominator is :       x 

      The right denominator is :       4 

        Number of times each prime factor
        appears in the factorization of: Prime 
 Factor  Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right} 2022 Product of all 
 Prime Factors 144

                  Number of times each Algebraic Factor
            appears in the factorization of:    Algebraic    
    Factor     Left 
 Denominator  Right 
 Denominator  L.C.M = Max 
 {Left,Right}  x 101


      Least Common Multiple: 
      4x 

Calculating Multipliers :

 8.3    Calculate multipliers for the two fractions 


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 4

   Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

 8.4      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent, y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well. 

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. (x-1) • (x-3) • 4 —————————————————— = ————————————————— L.C.M 4x R. Mult. • R. Num. 17 • x —————————————————— = —————— L.C.M 4x

Adding fractions that have a common denominator :

 8.5       Adding up the two equivalent fractions 

(x-1) • (x-3) • 4 - (17 • x) 4x2 - 33x + 12 ———————————————————————————— = —————————————— 4x 4x

Trying to factor by splitting the middle term

 8.6     Factoring  4x2 - 33x + 12 

The first term is,  4x2  its coefficient is  4 .
The middle term is,  -33x  its coefficient is  -33 .
The last term, "the constant", is  +12 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 12 = 48 

Step-2 : Find two factors of  48  whose sum equals the coefficient of the middle term, which is   -33 .

     -48   +   -1   =   -49     -24   +   -2   =   -26     -16   +   -3   =   -19     -12   +   -4   =   -16     -8   +   -6   =   -14     -6   +   -8   =   -14


For tidiness, printing of 14 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !! 
Conclusion : Trinomial can not be factored

Equation at the end of step  8  :

4x2 - 33x + 12 —————————————— = 0 4x

Step  9  :

When a fraction equals zero :

 9.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

4x2-33x+12 —————————— • 4x = 0 • 4x 4x

Now, on the left hand side, the  4x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   4x2-33x+12  = 0

Parabola, Finding the Vertex :

 9.2      Find the Vertex of   y = 4x2-33x+12

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 4 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   4.1250  

 Plugging into the parabola formula   4.1250  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 4.13 * 4.13 - 33.0 * 4.13 + 12.0 
or   y = -56.063

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x2-33x+12
Axis of Symmetry (dashed)  {x}={ 4.13} 
Vertex at  {x,y} = { 4.13,-56.06}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 0.38, 0.00} 
Root 2 at  {x,y} = { 7.87, 0.00} 

Solve Quadratic Equation by Completing The Square

 9.3     Solving   4x2-33x+12 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x2-(33/4)x+3 = 0

Subtract  3  from both side of the equation :
   x2-(33/4)x = -3

Now the clever bit: Take the coefficient of  x , which is  33/4 , divide by two, giving  33/8 , and finally square it giving  1089/64 

Add  1089/64  to both sides of the equation :
  On the right hand side we have :
   -3  +  1089/64    or,  (-3/1)+(1089/64) 
  The common denominator of the two fractions is  64   Adding  (-192/64)+(1089/64)  gives  897/64 
  So adding to both sides we finally get :
   x2-(33/4)x+(1089/64) = 897/64

Adding  1089/64  has completed the left hand side into a perfect square :
   x2-(33/4)x+(1089/64)  =
   (x-(33/) • (x-(33/)  =
  (x-(33/)2 
Things which are equal to the same thing are also equal to one another. Since
   x2-(33/4)x+(1089/64) = 897/64 and
   x2-(33/4)x+(1089/64) = (x-(33/)2 
then, according to the law of transitivity,
   (x-(33/)2 = 897/64

We'll refer to this Equation as  Eq. #9.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(33/)2   is
   (x-(33/)2/2 =
  (x-(33/)1 =
   x-(33/

Now, applying the Square Root Principle to  Eq. #9.3.1  we get:
   x-(33/ = √ 897/64 

Add  33/8  to both sides to obtain:
   x = 33/8 + √ 897/64 

Since a square root has two values, one positive and the other negative
   x2 - (33/4)x + 3 = 0
   has two solutions:
  x = 33/8 + √ 897/64 
   or
  x = 33/8 - √ 897/64 

Note that  √ 897/64 can be written as
  √ 897  / √ 64   which is √ 897  / 8 

Solve Quadratic Equation using the Quadratic Formula

 9.4     Solving    4x2-33x+12 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A 

  In our case,  A   =     4
                      B   =   -33
                      C   =   12 

Accordingly,  B2  -  4AC   =
                     1089 - 192 =
                     897

Applying the quadratic formula :

               33 ± √ 897 
   x  =    ——————
                      8

  √ 897   , rounded to 4 decimal digits, is  29.9500
 So now we are looking at:
           x  =  ( 33 ±  29.950 ) / 8

Two real solutions:

 x =(33+√897)/8= 7.869 

or:

 x =(33-√897)/8= 0.381 

Supplement : Solving Quadratic Equation Directly

Solving  x2-4x+3  = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 10.1      Find the Vertex of   y = x2-4x+3

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 1 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   2.0000  

 Plugging into the parabola formula   2.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 2.00 * 2.00 - 4.0 * 2.00 + 3.0 
or   y = -1.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2-4x+3
Axis of Symmetry (dashed)  {x}={ 2.00} 
Vertex at  {x,y} = { 2.00,-1.00}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = { 1.00, 0.00} 
Root 2 at  {x,y} = { 3.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 10.2     Solving   x2-4x+3 = 0 by Completing The Square .

 Subtract  3  from both side of the equation :
   x2-4x = -3

Now the clever bit: Take the coefficient of  x , which is  4 , divide by two, giving  2 , and finally square it giving  4 

Add  4  to both sides of the equation :
  On the right hand side we have :
   -3  +  4    or,  (-3/1)+(4/1) 
  The common denominator of the two fractions is  1   Adding  (-3/1)+(4/1)  gives  1/1 
  So adding to both sides we finally get :
   x2-4x+4 = 1

Adding  4  has completed the left hand side into a perfect square :
   x2-4x+4  =
   (x-2) • (x-2)  =
  (x-2)2 
Things which are equal to the same thing are also equal to one another. Since
   x2-4x+4 = 1 and
   x2-4x+4 = (x-2)2 
then, according to the law of transitivity,
   (x-2)2 = 1

We'll refer to this Equation as  Eq. #10.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-2)2   is
   (x-2)2/2 =
  (x-2)1 =
   x-2

Now, applying the Square Root Principle to  Eq. #10.2.1  we get:
   x-2 = √ 1 

Add  2  to both sides to obtain:
   x = 2 + √ 1 

Since a square root has two values, one positive and the other negative
   x2 - 4x + 3 = 0
   has two solutions:
  x = 2 + √ 1 
   or
  x = 2 - √ 1 

Solve Quadratic Equation using the Quadratic Formula

 10.3     Solving    x2-4x+3 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A 

  In our case,  A   =     1
                      B   =    -4
                      C   =   3 

Accordingly,  B2  -  4AC   =
                     16 - 12 =
                     4

Applying the quadratic formula :

               4 ± √ 4 
   x  =    ————
                   2

Can  √ 4 be simplified ?

Yes!   The prime factorization of  4   is
   2•2  
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 4   =  √ 2•2   =
                ±  2 • √ 1   =
                ±  2 

So now we are looking at:
           x  =  ( 4 ± 2) / 2

Two real solutions:

x =(4+√4)/2=2+= 3.000 

or:

x =(4-√4)/2=2-= 1.000 

Two solutions were found :

 x =(33-√897)/8= 0.381 x =(33+√897)/8= 7.869


Processing ends successfully