Quadratic Function: Find Zeros

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

Formula

Formula

y = a(x - r1)(x - r2)
is a quadratic function in factored form.

The zeros of the quadratic function are
r1, r2.

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
x2 - 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:
-2x2 + 8 = -2(x2 - 4).

Common Monomial Factor

(x2 - 4)
= (x2 - 22)
= (x + 2)(x - 2)

Factor the Difference of Two Squares: a2 - b2

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.