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Completing the Square Calculator
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The Standard Quadratic Equation is given by ax2 + bx + c = 0, to find the roots we can use two methods:

  1. By using Quadratic Formula
  2. By Completing the Square
Completing the Square Calculator (or Completing the Square Solver or Complete the Square Calculator) takes a quadratic equation and solves it by completing the square.

Below is given a default quadratic equation, click "Solve by completing the square calculator". You can find the value of x by making perfect square and then taking square root on both sides.
 

Steps to Completing the Square

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Step 1 :  

Observe the Coefficient of x2 and then divide all terms by it.



Step 2 :  

Move constant on right hand side and x terms on left hand side.



Step 3 :  

Now write the perfect square on left hand side.



Step 4 :  

Take the square root on both sides and solve for x.



Example Problems

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  1. Solve the equation 5x2 - 3x - 6 = 0 by completing the square method.


    Step 1 :  

    Given equation = 5x2 - 3x - 6 = 0
    Given coefficient of x2 = 5
    Divide the equation by 5
    $\frac{5x^{2}}{5}$ - $\frac{3x}{6}$ - $\frac{6}{3}$ = 0
    x2 - $\frac{3x}{6}$ - $\frac{6}{3}$ = 0



    Step 2 :  

    Move constant on right hand side and x terms on left hand side.
    x2 - $\frac{3x}{6}$ = $\frac{6}{3}$



    Step 3 :  

    Now take the half of the x-term and then square it, to make the perfect square on the left hand side.
    x2 - $\frac{3x}{6}$ + ($\frac{3}{10}$)2 = $\frac{6}{3}$ + ($\frac{3}{10}$)2
    (x - $\frac{3}{10}$)2 = $\frac{6}{5}$ + $\frac{9}{100}$
    (x - $\frac{3}{10}$)2 = $\frac{129}{100}$



    Step 4 :  

    Take the square root on both sides and solve for x.
    (x - $\frac{3}{10}$) = $\pm$ $\frac{129}{100}$
    x = $\frac{3}{10}$ $\pm$ $\frac{129}{100}$
    x = $\frac{1}{10}$(3 + $\sqrt{129}$)
    x = $\frac{1}{10}$(3 - $\sqrt{129}$)



    Answer  :  

    x = $\frac{1}{10}$(3 + $\sqrt{129}$)
       = $\frac{1}{10}$(3 - $\sqrt{129}$)



  2. Solve the equation x2 + 4x - 3 = 0 by completing the square method.


    Step 1 :  

    Given equation = x2 + 4x - 3 = 0
    Given coefficient of x2 = 1



    Step 2 :  

    Move constant on right hand side and x terms on left hand side.
    x2 + 4x - 3 = 0
    x2 + 4x = 3



    Step 3 :  

    Now take the half of the x-term and then square it, to make the perfect square on the left hand side.
    x2 + 4x + (2)2 = 3 + (2)2
    (x + 2)2 = 3 + 4
    (x + 2)2 = 7



    Step 4 :  

    Take the square root on both sides and solve for x.
    (x + 2) = $\pm$ $\sqrt{7}$
    x = -2 $\pm$ $\sqrt{7}$
    x = -2 + $\sqrt{7}$
    x = -2 - $\sqrt{7}$



    Answer  :  

    x = -2 + $\sqrt{7}$
       = -2 - $\sqrt{7}$



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