Problem of the Week

Updated at Sep 9, 2024 5:53 PM

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Mar 24, 2025

This week we have another algebra problem:

How can we compute the factors of $42{q}^{2}-60q+18$ ?

Let's start!

42{q}^{2}-60q+18

Find the Greatest Common Factor (GCF).

What is the largest number that divides evenly into

42{q}^{2}

-60q

, and

18

It is

6

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

42{q}^{2}

-60q

, and

18

It is 1, since

q

is not in every term.

Multiplying the results above,

The GCF is

6

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GCF =

6

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

6(\frac{42{q}^{2}}{6}+\frac{-60q}{6}+\frac{18}{6})

Simplify each term in parentheses.

6(7{q}^{2}-10q+3)

Split the second term in

7{q}^{2}-10q+3

into two terms.

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

7\times 3=21

Ask: Which two numbers add up to

-10

and multiply to

21

-3

and

-7

Split

-10q

as the sum of

-3q

and

-7q

7{q}^{2}-3q-7q+3

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

6(7{q}^{2}-3q-7q+3)

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

6(q(7q-3)-(7q-3))

Factor out the common term

7q-3

6(7q-3)(q-1)

Done