Solving Quadratic Equations: Completing the Square

Quadratic Equations and Completing the Square

Introduction to Quadratic Equations

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$. There are several methods to solve quadratic equations, including factoring, using the square root property, completing the square, and the quadratic formula.

Methods for Solving Quadratic Equations

Factoring	Square Root Property	Completing the Square	Other Methods
Use if the equation has obvious factors. Write in standard form $ax^2 + bx + c = 0$. Factor completely. Set each factor to zero and solve. Check solutions.	Use if the equation is in the form $(x + h)^2 = k$. Take the square root of both sides. Solve for $x$.	Use to rewrite $ax^2 + bx + c = 0$ as a perfect square trinomial. Add $(\frac{b}{2})^2$ to both sides. Factor the left side as $(x + \frac{b}{2})^2$. Solve using the square root property.	Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form $(x + h)^2 = k$.

• Start with $ax^2 + bx + c = 0$.

• If $a \neq 1$, divide both sides by $a$ to make the coefficient of $x^2$ equal to 1.

• Move the constant term to the other side: $x^2 + bx = -c$.

• Add $(\frac{b}{2})^2$ to both sides to create a perfect square trinomial.

• Factor the left side as $(x + \frac{b}{2})^2$.

• Solve for $x$ using the square root property.

Key Formula

To complete the square for $x^2 + bx$:

$x^2 + bx + \left(\frac{b}{2}\right)^2 = (x + \frac{b}{2})^2$

Square Root Property

If an equation is in the form $(x + h)^2 = k$, then:

$x + h = \pm \sqrt{k}$

Solve for $x$ by isolating the variable.

Example: Solving by Completing the Square

• Given: $x^2 + 6x = 7$

• Add $(\frac{6}{2})^2 = 9$ to both sides: $x^2 + 6x + 9 = 7 + 9$

• Rewrite: $(x + 3)^2 = 16$

• Take the square root: $x + 3 = \pm 4$

• Solutions: $x = -3 \pm 4$

• Therefore, $x = 1$ or $x = -7$

Practice Problems

• Solve $x^2 + 3x - 5 = 0$ by completing the square.

• Solve $3x^2 - 6x - 9 = 0$ by completing the square.

Steps for Completing the Square (Summary Box)

1. Simplify the equation to $x^2 + bx = c$.

2. Add $(\frac{b}{2})^2$ to both sides.

3. Factor the left side as $(x + \frac{b}{2})^2$.

4. Solve using the square root property.

Additional info: Completing the square is also useful for deriving the quadratic formula and for analyzing the vertex form of a quadratic function.