**x**, called

*roots*, that make this equation true (i.e., makes the left hand side equal to 0.) The formula below will find the two roots of the equation:There are two roots because the

**±**symbol means consider it as a

**+**one time and as a

**−**another time.

## How to Solve Quadratic Equations Using the Quadratic Formula

Solving a quadratic equation using the quadratic formula is easy.## Question

Solve the quadratic equation shown below using the quadratic formula.## Step-by-Step:

## 1

^{2}−5x + 2 = 0 ⇔ ax

^{2}+ bx + c = 0

**a**,

**b**and

**c**.

**a**= 2,

**b**= −5,

**c**= 2

## 2

## 3

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}$$

$$\:\:\:\: = \frac{5 \pm \sqrt{(-5 \times -5) - 4 \times 2 \times 2}}{2 \times 2}$$

$$\:\:\:\: = \frac{5 \pm \sqrt{25 - 16}}{4}$$

$$\:\:\:\: = \frac{5 \pm \sqrt{9}}{4}$$

$$\:\:\:\: = \frac{5 \pm 3}{4}$$

## 4

$$x = \frac{5 + 3}{4}$$

$$\:\:\:\: = \frac{8}{4}$$

$$\:\:\:\: = 8 \div 4$$

$$\mathbf{x = 2}$$

**x = 2**is a root of the quadratic equation.

## 5

$$x = \frac{5 - 3}{4}$$

$$\:\:\:\: = \frac{2}{4}$$

$$\mathbf{x = \frac{1}{2}}$$

**x =**is a root of the quadratic equation.

^{1}⁄_{2}## Answer:

We have solved the quadratic equation:**x =**,

^{1}⁄_{2}**x = 2**.

## 2 Roots

Quadratic equations have 2 roots, and the quadratic equation finds both of them. Look closely at the formula, and you'll see a ± sign:$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$$

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

## The Discriminant

The term in the formula that appears in a square root is called the*discriminant*:

*discrimina*tes between the 3 possible cases for the roots of a quadratic equation. We can visualize this by looking at a graph of a quadratic equation. The roots are the points where the curve crosses the horizontal x-axis.

**b**: there are 2 real, distinct roots.^{2}− 4ac > 0**b**: there is one repeated root.^{2}− 4ac = 0**b**: there are 2 complex roots.^{2}− 4ac < 0

## Beware

## Be Careful with Signs

**a**,

**b**and

**c**may be negative. Make sure you remember this when inserting them into the equation - write them inside brackets if need be.

## Interactive Widget

You can use this**interactive widget**to create a graph of your quadratic equation. Use the buttons to change the values of the quadratic equation.

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