# Quadratic Function: Find Zeros

How to find the zeros of a quadratic function: formula, 2 examples, and their solutions.

## Formula

### Formula

y = a(x - r_{1})(x - r_{2})

is a quadratic function in factored form.

The zeros of the quadratic function are

r_{1}, r_{2}.

The zeros are the x-intercepts of a function.

## Example 1

### Example

### Solution

To find the zeros of the quadratic function,

change the quadratic function to factored form.

Factor the right side

x^{2} - 2x - 3.

Find a pair of numbers

whose product is the constant term -3

and whose sum is the coefficient of the middle term -2.

-3⋅1 = -3

-3 + 1 = -2

So the given function becomes

y = (x - 3)(x + 1).

Factor a Quadratic Trinomial

Set (right side) = 0:

(x - 3)(x + 1) = 0.

Then the roots of this quadratic equation

are the zeros of the quadratic function.

Quadratic Equation: by Factoring

Case 1) x - 3 = 0

Then x = 3.

This is the zero for case 1.

Case 2) x + 1 = 0

Then x = -1.

This is the zero for case 2.

Case 1) x = 3

Case 2) x = -1

Then x = -1, 3.

So the zeros are

x = -1, 3.

### Graph

This is the graph of y = (x - 3)(x + 1).

The zeros are -1 and 3.

## Example 2

### Example

### Solution

To find the zeros of the quadratic function,

change the quadratic function to factored form.

Factor the right side:

-2x^{2} + 8 = -2(x^{2} - 4).

Common Monomial Factor

(x^{2} - 4)

= (x^{2} - 2^{2})

= (x + 2)(x - 2)

Factor the Difference of Two Squares: a^{2} - b^{2}

Set (right side) = 0:

-2(x + 2)(x - 2) = 0.

Then the roots of this quadratic equation

are the zeros of the quadratic function.

Quadratic Equation: by Factoring

Case 1) x + 2 = 0

Then x = -2.

This is the zero for case 1.

Case 2) x - 2 = 0

Then x = 2.

This is the zero for case 2.

Case 1) x = -2

Case 2) x = 2

Then x = ±2.

So the zeros are

x = ±2.

### Graph

This is the graph of y = -2(x + 2)(x - 2).

The zeros are ±2:

-2 and 2.