Lesson 10.2: Solve Quadratic Equations by Completing the Square

New Concept

Completing the square is a universal method for solving any quadratic equation. By algebraically creating a perfect square trinomial, you can transform the equation into the form $(x-k)^2 = d$ and solve it using the Square Root Property.

What’s next

Next, you'll master this technique through interactive examples and practice problems, starting with binomials and then solving full quadratic equations.

Property

To make a perfect square trinomial from an expression like $x^2 + bx$, we use the binomial squares pattern. The goal is to find a number to add that completes the pattern $a^2 + 2ab + b^2 = (a+b)^2$.

To complete the square of $x^2 + bx$:
Step 1. Identify $b$, the coefficient of $x$.
Step 2. Find $(\frac{1}{2}b)^2$, the number to complete the square.
Step 3. Add the $(\frac{1}{2}b)^2$ to $x^2 + bx$. The result is $x^2 + bx + (\frac{1}{2}b)^2$, which factors to $(x + \frac{b}{2})^2$.

Examples

• To complete the square for $x^2 + 12x$, we find $(\frac{1}{2} \cdot 12)^2 = 6^2 = 36$. The perfect square trinomial is $x^2 + 12x + 36$, which factors to $(x+6)^2$.

Property

When solving an equation, you must perform the same operation on both sides. When completing the square, the term you add to create a perfect square trinomial on one side must also be added to the other side.

To solve a quadratic equation of the form $x^2 + bx + c = 0$:
Step 1. Isolate the variable terms on one side and the constant terms on the other.
Step 2. Find $(\frac{1}{2} \cdot b)^2$ and add it to both sides of the equation.
Step 3. Factor the perfect square trinomial as a binomial square.
Step 4. Use the Square Root Property.
Step 5. Simplify the radical and solve the two resulting equations.

Examples

• To solve $x^2 + 6x - 7 = 0$, first isolate the constant: $x^2 + 6x = 7$. Add $(\frac{6}{2})^2 = 9$ to both sides: $x^2 + 6x + 9 = 16$. Factor and solve: $(x+3)^2 = 16$, so $x+3 = \pm 4$, which gives $x=1$ and $x=-7$.

Property

The process of completing the square works best when the leading coefficient is one. If the $x^2$ term has a coefficient $a$ other than 1, you must first take a preliminary step to make the coefficient equal to one.

To solve an equation of the form $ax^2 + bx + c = 0$:
Step 1. Divide both sides of the equation by the leading coefficient, $a$. This gives an equation of the form $x^2 + (b/a)x + (c/a) = 0$.
Step 2. Proceed with the standard steps for completing the square.

Examples

• To solve $2x^2 + 8x - 10 = 0$, first divide by 2 to get $x^2 + 4x - 5 = 0$. Then solve by completing the square: $x^2 + 4x = 5 \implies (x+2)^2 = 9 \implies x+2 = \pm 3$. The solutions are $x=1$ and $x=-5$.