Lesson 4: Solving Quadratic Equations by Completing the Square

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 $x=-7$.