QuestionFully factorise the expression $15 n^{2}-16 n p-15 p^{2}$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to check your answer by expanding it back out!

STEP 2

1. Factor the expression

STEP 3

We're dealing with a quadratic expression in two variables, $n$ and $p$ .
Let's look closely at our expression: $15n^2 - 16np - 15p^2$ .

STEP 4

We're going to use a clever trick!
We'll treat this like a quadratic in $n$ , where $p$ is just hanging out along for the ride.
Think of it like this: $15n^2 - (16p)n - 15p^2$ .
See how we grouped the $16p$ with the $n$ ?

STEP 5

We need two numbers that multiply to $(15) \cdot (-15p^2) = -225p^2$ and add up to $-16p$ .
Now, think carefully... $9p$ and $-25p$ do the trick! $9p \cdot -25p = -225p^2$ and $9p + (-25p) = -16p$ .
Perfect!

STEP 6

Let's rewrite our middle term using these magic numbers: $15n^2 + 9np - 25np - 15p^2$ .

STEP 7

Now, we'll group the terms: $(15n^2 + 9np) + (-25np - 15p^2)$ .

STEP 8

From the first group, we can factor out $3n$ : $3n(5n + 3p)$ .
From the second group, we can factor out $-5p$ : $-5p(5n + 3p)$ .
Notice how we have a common factor of $(5n + 3p)$ !

STEP 9

Now, we can factor out the common factor $(5n + 3p)$ : $(5n + 3p)(3n - 5p)$ .
Boom! We did it!

STEP 10

The fully factored expression is $(5n + 3p)(3n - 5p)$ .

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