# Complete The Square - Calculator

An online calculator that completes the square of a quadratic expression.

## Complete the Square of a Quadratic Expression

Completing the square is to write a quadratic expression of the form $a x^2 + b x + c$ into the form $a (x + h)^2 + k$.
step 1: Factor coefficient $a$ out of the terms in $x$
$ax^2 + b x + c = a( x^2 + \dfrac{b}{a} x )+ c$
step 2: Add and subtract $(b/2a)^2$ inside the parentheses (it is like adding zero! and we therefore do no change the expression).
$= a( x^2 + \dfrac{b}{a} x + (\dfrac{b}{2a})^2 - (\dfrac{b}{2a})^2 )+ c$
Group the first three terms inside the parentheses.
$= a( x^2 + \dfrac{b}{a} x + (\dfrac{b}{2a})^2) - a (\dfrac{b}{2a})^2 + c$
The terms inside the parentheses are grouped as a square hence "completing the square".
$= a( x + \dfrac{b}{2a})^2 - a (\dfrac{b}{2a})^2 + c$
The above may be written as
$= a(x+h)^2 + k$
with $h = \dfrac{b}{2a}$ and $k = c - a (\dfrac{b}{2a})^2$

Example 1: Complete the square in the expression $2x^2 + 6 x + 5/2$
Solution
Factor the coefficient of $x^2$ out of the terms in x.
$2x^2 + 6 x - 1 = 2(x^2 + 3) + 5/2$
add and subtract $(3/2)^2$ inside the parentheses
$= 2(x^2 + 3 x + (3/2)^2 - (3/2)^2) + 5/2$
Group the first three terms inside the parentheses
$= 2(x^2 + 3 x + (3/2)^2) - 2(3/2)^2 + 5/2$
Write the terms inside the parentheses as a square and group the constant terms outside the parentheses
$= 2(x + 3/2)^2 - 2$

## Complete the Square Calculator

1 - Enter the coefficients a, b and c and press "Complete Square". a, b and c can be entered as integers, fractions or numbers with a decimal point as shown below. This calculator helps you practice on as many examples as you wish and check your answers.

 2 $x^2 +$6$x +$5/2

## Applications of Completing the Square

The method of completing the square may be applied in the following situation
Example solve: $2x^2 + 6 x + 5/2 = 0$
We have completed the square in example 1 for the expression $2x^2 + 6 x + 5/2 = 2(x + 3/2)^2 - 2$ and therefore solving the given equation is equivalent to solving the equation
$2(x + 3/2)^2 - 2 = 0$
$(x + 3/2)^2 = 1$
$x + 3/2 = \pm \sqrt 1$
$x = - 3/2 \pm 1$ , 2 solutions : $x_1 = -5/2$ , $x_2 = - 1/2$
2) Find the vertex and x-intercepts of the graph of a quadratic function.
Example: Find the vertex and the x-intercepts of the graph of the function $y = 2x^2 + 6 x + 5/2$
Use the equivalent expression obtained above
$y = 2x^2 + 6 x + 5/2 = 2(x + 3/2)^2 - 2 = a(x + h) + k$
The vertex is at the point $(- h , k) = (- 3/2 , -2)$
The x-intercepts are the solutions of the equation $2x^2 + 6 x + 5/2 = 0$ which has been solved in example 1.
The graph below is that of the function $y = 2x^2 + 6 x + 5/2$ with the x-intercepts and the vertex.

3) The third application of completing the square is in calculus where integrals involve quadratic expressions are evaluated.