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All courses Algebra I · F-IF.8.a 69 of 76
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

Apply the definition of a zero

\[\begin{aligned} f(x)=0 \\ (x-2)(x+5)=0 \end{aligned}\]

Zeros are inputs where the function output equals zero, so set the factored expression equal to zero.

02

Use the zero-product property

\[x-2=0~\text{or}~x+5=0\]

A product is zero exactly when at least one factor is zero. This is why factored form exposes the roots directly.

03

Solve each factor equation

\[\begin{aligned} x-2=0~->~x=2 \\ x+5=0~->~x=-5 \end{aligned}\]

Undoing the constants gives the two distinct zero inputs.

04

State and verify both zeros

\[\begin{aligned} \text{zeros}~x=2~\text{and}~x=-5 \\ f(2)=0(7)=0 \\ f(-5)=(-7)(0)=0 \end{aligned}\]

Substitution makes one factor zero at each input, confirming both answers.

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Another way

  1. Use the factor-root correspondence: x-r gives root r, so x-2 gives 2 and x+5=x-(-5) gives -5.

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Common mistake

Do not report x=5 from the factor x+5. Solving x+5=0 requires subtracting 5, giving x=-5.

Solution walkthrough video