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All courses Math III · A-CED.1 8 of 55
Create and solve one-variable equations and inequalities from contexts using all studied expression types, including simple root functions.

Write and solve a polynomial equation from a context

Problem
Write and solve a polynomial equation for this context: a box has volume x(x + 2)(x + 3) = 60.
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Hint

Move everything to one side so the volume equation becomes a polynomial equation equal to \(0\).

A box dimension must be positive, so after solving the polynomial equation you keep only the positive root that fits the context.

Solution walkthrough

01

Write the polynomial equation

\[x(x+2)(x+3)-60~=~0\]

The context already gives the volume model, so subtract \(60\) from both sides to write it in standard polynomial form.

02

Expand the product

\[x(x^2+5x+6)-60~=~0\;\Rightarrow\;x^3+5x^2+6x-60~=~0\]

Multiply the factors to see the cubic equation clearly.

03

Solve for the real root

\[x~\approx~2.462\]

This cubic does not factor nicely over the integers, so solve numerically to find the real positive solution.

04

Interpret the solution

\[x~\approx~2.462\]

Because \(x\) is a box dimension, the positive root is the meaningful answer.

05

State the final answer

\[\text{The}~\text{positive}~\text{solution}~\text{is}~x~\approx~2.462,~\text{so}~\text{the}~\text{box}~\text{dimension}~\text{represented}~\text{by}~x~\text{is}~\text{about}~2.462.\]

The positive solution is \(x\approx 2.462\), so the box dimension represented by \(x\) is about \(2.462\).

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Another way

  1. You can graph \(y=x(x+2)(x+3)-60\) and find the positive x-intercept.

!

Common mistake

A common mistake is to stop at \(x(x+2)(x+3)=60\) without rewriting it as a polynomial equation equal to \(0\), which makes solving harder.