QuestionSimplify, $\frac{x^{2}+2 x-35}{-4 x^{2}-16 x+84}$

Studdy Solution

STEP 1

What is this asking? We're asked to simplify a funky fraction with x by factoring and then canceling out common factors. Watch out! Don't forget to factor out the negative sign in the denominator!
It's a sneaky little fella that can easily trip you up.

STEP 2

1. Factor the numerator
2. Factor the denominator
3. Simplify the fraction

STEP 3

Alright, let's factor that numerator, which is $x^{2} + 2x - 35$ .
We're looking for two numbers that add up to $+2$ and multiply to $-35$ .
Hmm, what could those be?

STEP 4

Aha! $+7$ and $-5$ fit the bill perfectly!
So, we can rewrite the numerator as $(x + 7) \cdot (x - 5)$ .
Boom!

STEP 5

Now, let's tackle the denominator: $-4x^{2} - 16x + 84$ .
First, let's factor out $-4$ to make things a bit easier.
This gives us $-4 \cdot (x^{2} + 4x - 21)$ .

STEP 6

Now we need two numbers that add up to $+4$ and multiply to $-21$ .
How about $+7$ and $-3$ ?
Perfect! So, the denominator becomes $-4 \cdot (x + 7) \cdot (x - 3)$ .
Nice!

STEP 7

Now, let's put it all together.
Our fraction is now $\frac{(x + 7) \cdot (x - 5)}{-4 \cdot (x + 7) \cdot (x - 3)}$

STEP 8

We can divide both the numerator and the denominator by $(x + 7)$ .
Remember, dividing something by itself is like dividing by one!
This leaves us with: $\frac{x - 5}{-4 \cdot (x - 3)}$

STEP 9

And there we have it, our simplified fraction!
We can also write it as $\frac{-(x - 5)}{4 \cdot (x - 3)}$ or $\frac{5 - x}{4 \cdot (x - 3)}$ Magnificent!

STEP 10

Our simplified fraction is $\frac{x - 5}{-4 \cdot (x - 3)}$ or $\frac{5 - x}{4 \cdot (x - 3)}$ .

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