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.
## Applications of Completing the Square
The method of completing the square may be applied in the following situation
1) to solve quadratic equations.
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.
### More References and linksCompleting the Square of Quadratic Expressions
Step by Step Solver Calculator to Complete the Square
Evaluate Integrals Involving Quadratic Expressions
Using Completing Square |