All courses Math II · A-SSE.3.a 11 of 73
Factor quadratics to reveal zeros of the function they define.

Factor a monic quadratic to reveal zeros

Problem
Factor monic quadratic x²−7x+12 and reveal its zeros. Enter product/sum targets, integer pair, factored form, all zeros, and coefficient/root check.
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Hint

List integer factor pairs of c and test their sums.

For x²+bx+c, factor numbers multiply to c and add to b; x−r=0 gives zero r.

Solution walkthrough

01

Find the factor pair

\[(-3)(-4)~=~12~\text{and}~-3+(-4)~=~-7\]

These numbers match both the product and the sum conditions.

02

Write the factored form

\[x^2-7x+12~=~(x-3)(x-4)\]

The numbers \(-3\) and \(-4\) become the binomial constants.

03

Set each factor equal to zero

\[x-3~=~0~\text{or}~x-4~=~0\]

A product is zero when at least one factor is zero.

04

State the factorization and zeros

\[\text{target}~\text{product}=12;~\text{target}~\text{sum}=-7;~\text{integer}~\text{pair}=-3,-4;~\text{factored}=(x−3)(x−4);~\text{zeros}=x=3,4;~\text{check}=\text{sum}~\text{roots}7~\text{gives}~\text{coefficient}-7~\text{and}~\text{product}12\]

So the requested factorization is \((x-3)(x-4)\), and the zeros are \(3\) and \(4\).

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Another way

  1. Expand \((x-3)(x-4)\) to check that it gives \(x^2-7x+12\).

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Common mistake

Choosing numbers that multiply to \(12\) but add to \(7\) instead of \(-7\).