Solving Quadratic Equations and Inequalities

Completing the Square

All quadratic equations can be solved by completing the square.
Another method used to solve quadratic equations is a process called completing the square. When completing the square, a constant term is added to both sides of a quadratic equation so that a perfect square trinomial is formed. Then, to solve the equation, take the square root of both sides of the equation. Give exact answers using radicals (square root symbols) unless the instructions say to give approximate answers. Approximate answers may also help when drawing a sketch of the related function.
Step-By-Step Example
Complete the Square to Solve a Quadratic Equation
Solve the quadratic equation:
2x28x14=02x^2-8x-14=0
Step 1

When the leading coefficient is not 1, divide each term by the leading coefficient.

The leading coefficient in the equation is 2. So, divide each term by 2.
2x228x2142=02x24x7=0\begin{aligned}\frac{2x^2}{2}-\frac{8x}{2}-\frac{14}{2}&=\frac{0}{2}\\x^2-4x-7&=0\end{aligned}
Step 2
Write the equation in the form:
x2+bx=cx24x=7\begin{aligned}x^2+bx&=c\\x^2-4x&=7\end{aligned}
Step 3
Determine the value that is added to both sides of the equation to make the left side a perfect square trinomial.
(b2)2=(42)2=(2)2=4\begin{aligned}\left (\frac{b}{2} \right )^{2}&=\left (\frac{-4}{2}\right)^{2}\\&=(-2)^{2}\\&=4\end{aligned}
Step 4
Add 4 on the left side of the given equation to complete the square. Add 4 to the other side to keep the equation balanced.
x24x+4=7+4x24x+4=11\begin{aligned}x^2-4x+4&=7+4\\x^2-4x+4&=11\end{aligned}
Step 5
Factor the perfect square trinomial.
(x2)2=11(x-2)^2=11
Step 6
Take the square root of both sides of the equation.
x2=±11x-2=\pm \sqrt{11}
Solution
Use both the positive and negative square root to write and solve two equations.
x2=11x=11+2orx2=11x=11+2\begin{aligned}x-2&=\sqrt{11}\\x&=\sqrt{11}+2\end{aligned}\;\;\;\;\;\text{or}\;\;\;\;\;\begin{aligned}x-2&=-\sqrt{11}\\x&=-\sqrt{11}+2\end{aligned}
The approximate solutions are:
x=11+23.3+25.3orx=11+23.3+21.3\begin{aligned}x&=\sqrt{11}+2\\&\approx 3.3+2\\&\approx5.3\end{aligned}\;\;\;\;\;\text{or}\;\;\;\;\;\begin{aligned}x&=-\sqrt{11}+2\\&\approx -3.3+2\\&\approx-1.3\end{aligned}