## Introduction

**Quadratic functions** are easy to recognize. The **polynomial expression** known as a *quadratic* contains a variable that is squared making it a **2nd degree equation**, and the graph is U-shaped. Quadratic expressions that are equal to zero are called **quadratic equations**.

The **standard form** of a quadratic equation is: $ax^{2} + bx + c = 0$

The graph of a quadratic equation has a recognizable shape – **a parabola**. The parabola may open up or down, and the direction of the opening is determined by the sign of the leading **coeeficient**.

## Solving Quadratic Equations by Factoring

To find the **solutions to quadratic equations**, also known as the **zeros** or **roots**, set the quadratic expression equal to zero then **factor**. The values for x identify where the graph touches the x-axis. There are several methods you can use to factor quadratic equations.

### Greatest Common Factor

Identify the **greatest common factor** (**GCF**) of all the terms in the quadratic expression and use the reverse of the **Distributive Property** to factor.

This equation has a GCF equal to 2x.

$\begin{align} 2x^{2} + 2x& = 0\\ 2x(x + 2) &= 0 \end{align}$

x = -2, 0

The graph touches the x-axis at -1 and 0.

### Square Root Property

Use the **property of square roots** to find the zeros of quadratic equations such as this one.

$\begin{align} x^{2} - 36 &= 0\\ x^{2} - 36 + 36 &= 0 +36\\ x^{2} &= 36\\ x&=\sqrt{36}\\ \sqrt{36} &=\pm 6 \end{align}$

The solution to the quadratic equation is -6 and 6.

### Foil

The **foil method** is used to simplify the product of two binomials, so the reverse of the foil method can be used to factor quadratic equations with trinomial expressions having a equal to 1.

To reverse the foil method, find factors of the c that sum to the b.

$ax^{2} + bx + c = 0$

$x^{2} + 7x + 6 = 0$

For the product of 6, the factors 1 and 6 sum to 7. Inside two sets of parentheses, add the constants of 6 and 1 to x respectively then set each binomial equal to zero and solve to determine the roots of the equations.

$(x + 6)(x+ 1) = 0$

$x = -6, -1$

To check your work, FOIL.

$\begin{align} (x + 6)(x+ 1)&=0\\ x^{2} + 6x + 1x +6&=0\\ x^{2} + 7x +6&=0 \end{align}$

The roots of the equation are -6 and -1.

### Difference of Two Squares

A quadratic equation that is the difference of two squares is also known as a **DOTS equation**. If you can recognize which quadratic equations are DOTS (difference of two squares), you can save yourself time when factoring quadratic equations.

To identify DOTS, look for a specific pattern in the quadratic equation. Notice there are only two terms and both are perfect squares. The solution is the square root of the constant.

$ax^{2} + bx + c = 0$

$\begin{align} x^{2} - 49 & =0\\ (x +7) (x -7)&=0 \end{align}$

The solution to this DOTS equations is -7 and 7.

### Factor by Grouping

When the quadratic equation has a **trinomial expression** with $a\neq 1$, you can **factor by grouping**. There are several steps to this method.

$ax^{2} + bx + c = 0$

$2x^{2} + -6x -8 =0$

Factor by Grouping

- Find the product of a and c.
- Identify two factors that sum to b.
- Write new values for bx.
- Group the factors using parentheses
- Factor out the GCF of each group
- Set up the binomial factors.

For this equation $a\times c = -16$. -2 and 8 sum to -6. Take a look at the next steps to solve this quadratic equation.

$\begin{align} 2x^{2} -6x -8 & =0\\ 2x^{2} +2x -8x -8 & =0\\ (2x^{2} +2x)+ (-8x - 8)&=0\\ 2x(x +1) -8(x +1)&=0\\ (2x -8)(x +1)&=0 \end{align}$

The solution to the equation is -1 and 4.

## Solving Quadratic Equations by Completing the Square

When you are unable to determine factors, you can use the **complete the square method** to solve quadratic equations. To determine the roots using this method, there are several steps.

$ax^{2} + bx + c = 0$

- Use the opposite operation to move the constant to the other side of the standard form.
- Take half of b, square it and add to both sides of the equation.
- Factor the perfect square on left side of the equation.
- Apply the square root property to solve.

$\begin{align} x^{2} + 2x -7 &= 0\\ x^{2} + 2x -7 +7&= 0 +7\\ x^{2} + 2x &=7\\ x^{2} + 2x + 1 &=7 + 1\\ x^{2} + 2x + 1 &=8\\ (x + 1)^{2}&= 8\\ x + 1 &= \pm\sqrt{8}\\ x + 1 -1&=-1 \pm\sqrt{8}\\ x&= -1\pm\sqrt{8} \end{align}$

The solution is x&= -1\pm\sqrt{8} which is -3.8 and 1.8.

## Solving Quadratic Equations with the Quadratic Formula

If there is no way to factor a quadratic equation or you simply prefer, you can always use the **quadratic formula** to determine the value(s) of x.

$ax^{2} + bx + c = 0$

Use the quadratic formula to solve this equation but first use the **discriminant** to learn about the roots.

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

### Using and Understanding the Discriminant

The **discriminant** is the **value under the radical**, and it provides valuable information about the roots of an quadratic equation.

Discriminant ${b^{2}-4ac}$

- if > 0 then there are two real roots
- if = 0 there is one root repeated
- if < 0 there are two complex roots

For this problem, the discriminant is greater than zero, so there are two real roots.

$x^{2} -6x -4 =0$

$\begin{align} x&=\frac{6\pm\sqrt{36+16}}{2}\\ x&=\frac{6\pm\sqrt{52}}{2}\\ x &= \frac{6}{2}\pm\frac{\sqrt{52}}{2} x&=3\pm 3.6\\ x&= -0.6, 6.6 \end{align}$

The roots for this equation are -0.6 and 6.6.