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All courses Math II · F-IF.8.a 23 of 73
Use factoring and completing the square to reveal zeros, extrema, and symmetry of quadratics.

Identify the zeros of a quadratic from factored form

Problem
Use factored form f(x)=(x-2)(x+5) to identify zeros.
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Hint

Set each factor equal to zero: \(x-2=0\) and \(x+5=0\).

In factored form, zeros occur when any factor equals zero.

Solution walkthrough

01

Write the factored function

\[f(x)~=~(x-2)(x+5)\]

The function is already factored, so the zeros can be read from the factors.

02

Set the first factor equal to zero

\[x-2~=~0~\text{so}~x~=~2\]

If this factor is zero, the whole product becomes zero.

03

Set the second factor equal to zero

\[x+5~=~0~\text{so}~x~=~-5\]

This gives the other input that makes the product zero.

04

State the zeros

\[\text{zeros}:~2,~-5\]

These are the x-values where the function equals zero.

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Another way

  1. Use the zero-product idea directly: a product is zero when at least one factor is zero.

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Common mistake

Changing the sign the wrong way and saying the zeros are \(-2\) and \(5\) instead of solving each factor carefully.