## Factoring

To solve a quadratic equation by factoring, follow these steps:

1. Move all non‐zero terms to the left side of the equation, effectively setting the polynomial equal to 0.

2. Factor the quadratic completely.

3. Set each factor equal to 0 and solve the smaller equations.

4. Plug each answer into the original equation to ensure that it makes the equation true.

**Example 1:** Solve the equation.

Add 13 *x* ^{2}and −10 *x* to both sides of the equation:

Factor the polynomial, set each factor equal to 0, and solve.

Because all three of these *x*‐values make the quadratic equation true, they are all solutions.

## The quadratic formula

If an equation can be written in the form *ax* ^{2} + *bx* + *c* = 0, then the solutions to that equation can be found using the **quadratic formula**:

This method is especially useful if the quadratic equation is not factorable. A word of warning: Make sure that the quadratic equation you are trying to solve is set equal to 0 before plugging the quadratic equation's coefficients *a*, *b*, and *c* into the formula. You should memorize the quadratic formula if you haven't done so already.

**Example 2:** Solve the quadratic equation.

Set the equation equal to 0:

The coefficients for the quadratic formula are *a* = −4, *b* = 6, and *c* = −1:

You can also write the answers as , the result of multiplying the numerators and denominators of both by −1. Note that the quadratic formula technique can easily find irrational and imaginary roots, unlike the factoring method.

## Completing the square

The most complicated, though itself not very difficult, technique for solving quadratic equations works by forcibly creating a trinomial that's a perfect square (hence the name). Here are the steps to follow:

1. Put the equation in form

ax^{2}+bx=c. In other words, move only the constant term to the right side of the equation.2. If

a≠ 1, divide the entire equation bya.3. Add the constant value to both sides of the equation.

4. Write the left side of the equation as a perfect square.

5. Take the square roots of both sides of the equation, remembering to add the “±” symbol on the right side.

6. Solve for

x.

**Example 3:** Solve the quadratic equation by completing the square.

Move the constant so it alone is on the right side:

Divide everything by the leading coefficient, since it's not 1:

Half of the *x*‐term's coefficient squared, . Add that value to both sides of the equation:

The left side is a perfect square:

Solve for *x*: Don't forget that you must include a ± sign when square rooting both sides of any equation.

The answer can also be written as , if rationalized.