Complete the Square Calculator
Completing the Square: Explanation
Completing the square rewrites a quadratic expression of the form
\[
ax^2 + bx + c
\]
into the vertex form
\[
a(x + h)^2 + k
\]
Step 1: Factor the coefficient of \(x^2\)
\[
ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c
\]
Step 2: Add and subtract \(\left(\frac{b}{2a}\right)^2\)
\[
= a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2
- \left(\frac{b}{2a}\right)^2 \right) + c
\]
Step 3: Group the perfect square
\[
= a\left(x + \frac{b}{2a}\right)^2
- a\left(\frac{b}{2a}\right)^2 + c
\]
Therefore,
\[
a(x + h)^2 + k
\]
where
\( h = \frac{b}{2a} \) and
\( k = c - a\left(\frac{b}{2a}\right)^2 \).
Example: Completing the Square
Complete the square for:
\[
2x^2 + 6x + \frac{5}{2}
\]
Factor the coefficient of \(x^2\):
\[
= 2(x^2 + 3x) + \frac{5}{2}
\]
Add and subtract \(\left(\frac{3}{2}\right)^2\):
\[
= 2\left(x^2 + 3x + \left(\frac{3}{2}\right)^2
- \left(\frac{3}{2}\right)^2 \right) + \frac{5}{2}
\]
Group the perfect square:
\[
= 2\left(x + \frac{3}{2}\right)^2 - 2
\]
Complete the Square Calculator
Enter the coefficients \(a\), \(b\), and \(c\) (integers, fractions, or decimals),
then press Complete Square.
Applications of Completing the Square
1. Solving Quadratic Equations
\[
2x^2 + 6x + \frac{5}{2} = 0
\]
\[
2\left(x + \frac{3}{2}\right)^2 - 2 = 0
\]
\[
(x + \frac{3}{2})^2 = 1
\]
\[
x = -\frac{3}{2} \pm 1
\]
Solutions:
\(x_1 = -\frac{5}{2}\),
\(x_2 = -\frac{1}{2}\)
2. Finding the Vertex and x-Intercepts
\[
y = 2(x + \frac{3}{2})^2 - 2
\]
Vertex:
\((-\frac{3}{2}, -2)\)
3. Applications in Calculus
Completing the square is frequently used when evaluating
integrals involving quadratic expressions
.