Problem of the Week

Updated at Mar 24, 2025 8:33 AM

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Mar 24, 2025

This week we have another algebra problem:

How can we factor $36{u}^{2}-54u+18$ ?

Let's start!

36{u}^{2}-54u+18

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

36{u}^{2}

-54u

, and

18

It is

18

What is the highest degree of $u$ that divides evenly into

36{u}^{2}

-54u

, and

18

It is 1, since

u

is not in every term.

Multiplying the results above,

The GCF is

18

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GCF =

18

Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)

18(\frac{36{u}^{2}}{18}+\frac{-54u}{18}+\frac{18}{18})

Simplify each term in parentheses.

18(2{u}^{2}-3u+1)

Split the second term in

2{u}^{2}-3u+1

into two terms.

Multiply the coefficient of the first term by the constant term.

2\times 1=2

Ask: Which two numbers add up to

-3

and multiply to

2

-1

and

-2

Split

-3u

as the sum of

-u

and

-2u

2{u}^{2}-u-2u+1

To get access to all 'How?' and 'Why?' steps, join Cymath Plus!

18(2{u}^{2}-u-2u+1)

Factor out common terms in the first two terms, then in the last two terms.

18(u(2u-1)-(2u-1))

Factor out the common term

2u-1

18(2u-1)(u-1)

Done