Question

How do you factor \[3{{x}^{2}}+45x+162\]?

Answer 1

Hint: In order to find the solution of the given question that is to factor the quadratic expression \[3{{x}^{2}}+45x+162\] use the sum-product pattern that is using splitting the middle term. Then common factors from two pairs and rewrite them in factored form.

Complete step by step solution:
According to the question, given expression in the question is as follows:
\[3{{x}^{2}}+45x+162\]
We can rewrite the above expression as follows:
\[\Rightarrow 3\left( {{x}^{2}}+15x+54 \right)\]
We will factorise the above expression by using the splitting the middle term method or we can sum-product pattern, we will have to find factors of \[1\] and \[54\] which combine and add [because of positive \[54\].] to give \[15\].
The signs in the brackets will be the same (because of positive \[9\]) and they will both be positive (because of positive\[15\].)
So, find the factors of \[1\] and \[54\] then cross multiply, we will have:
Factors of \[1\]: \[1\].
Factors of \[54\]: \[6\] and \[9\].
Here we see that \[1\times 6=6\] and \[1\times 9=9\]. And \[6+9=15\]. This means we can split \[15x\] into \[6x+9x\].
\[\Rightarrow 3\left( {{x}^{2}}+6x+9x+54 \right)\]
After the above step take the terms in common from the two pairs, we will have:
\[\Rightarrow 3\left( x\left( x+6 \right)+9\left( x+6 \right) \right)\]
Now rewrite the above expression in the factored-form, we get the final answer as:
\[\Rightarrow 3\left( x+6 \right)\left( x+9 \right)\]

Therefore, the factor-form of the given expression \[3{{x}^{2}}+45x+162\] is \[3\left( x+6 \right)\left( x+9 \right)\].

Note: There is a clue in the fact that \[15\] is odd. This tells us that \[54\] can neither be split as \[27\times 2\] nor as \[18\times 3\] because then both the factors of \[54\] should be positive numbers and after adding them it should give \[15\]. Therefore \[54\] has to be used as \[6\times 9\].