Top

Completing the Square Calculator
Top

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

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

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

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 - $\sqrt{129}$)

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 - $\sqrt{7}$

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