Question

Factor using the ‘ac’ method:

$18w^2-39w+18$

Short answer

The factors of$18w^2-39w+18$are$3(3w-2)(2w-3)$.

Step-by-step solution

Step 1. Given and explanation.

We have the equation $18w^2-39w+18$.

We will first take out any common factor, if present, from the equation to reduce it to the simplest form.

Then we will find any two numbers whose product comes out to be the product of first and last term and the sum results in the middle term.

Then by splitting the middle term and writing it into the an equation consisting of those two numbers, we will get the factors by taking the common factors put from that particular equation.

Step 2. Finding the two numbers. 

We can see that there is a common factor in the equation. So, taking it out will give us $3(6w^2-13w+6)$.

Now we will look for two numbers whose product comes out to be the product of first and last term, i.e. 36 and sum comes out to be the middle term, i.e., $-13$. We need to take negative factors because the sign of the middle term is negative. The factors of $36$and their sums are-

Numbers	Sums
role="math" localid="1644582076306" $-1and-36$	role="math" localid="1644582086195" $-37$
role="math" localid="1644582095800" $-2and-18$	role="math" localid="1644582105533" $-20$
role="math" localid="1644582117803" $-3and-12$	role="math" localid="1644582124949" $-15$
role="math" localid="1644582135433" $-4and-9$	$-13$
$-6and-6$	$-12$

Through this we can say that the two numbers will be$-4and-9$.

Step 3. Splitting the middle term and finding factors. 

Two numbers which we got are$-4,-9$.

Splitting the middle term gives us,

$$\begin{gathered} =3\left[6w^2-13w+6\right] \\ =3\left[6w^2-9w-4w+6\right] \\ =3\left[3w(2w-3)-2(2w-3)\right] \\ =3(3w-2)(2w-3) \end{gathered}$$

This shows that factors are$3(3w-2)(2w-3)$.

Step 5. Checking the solution.

We will check the solution by multiplying. If we get the same given equation, our calculations are right.

So,

$$\begin{gathered} =3(3w-2)(2w-3) \\ =\left[9w-6\right](2w-3) \\ =18w^2-27w-12w+18 \\ =18w^2-39w+18 \end{gathered}$$

Thus our calculations are right.