Question

Change the fraction w-3/w+5 into an equivalent fraction with the denominator w²+w=20. Choose the correct option
A. w²+7w-12/w²+2-20
B. w²-7w+12/w²+w-20
C. w²-7w-12/w²+w-20
D. w²+7w+12/w²+w-20

Answer 1

Answer:

$\sf B) \dfrac{ w^2 -7w +12 }{w^2 + w -2 0}$

Step-by-step explanation:

Let's find a common denominator for the given fraction $\sf \dfrac{w-3}{w+5}$ and the target denominator $\sf w^2 + w = 20$.

$\sf w^2 + w - 20 = 0$.

The target denominator can be factored as:

$\sf w^2 + w - 20$

$\sf w^2 (5-4)w - 20$

$\sf w^2 + 5w-4w - 20$

$\sf w(w+5)-4(w+5)$

$\sf (w+5)(w-4)$.

Now, we need to express the given fraction with the common denominator:

$\sf \dfrac{w-3}{w+5} \times \dfrac{w-4}{w-4}$

This results in:

$\sf \dfrac{(w-3)(w-4)}{(w+5)(w-4)}$

Distribute the parenthesis:

$\sf \dfrac{ w(w-4)-3(w-4)}{w(w-4)+5(w-4)}$

$\sf \dfrac{ w^2 - 4w - 3w +12 }{w^2 - 4w +5w - 20}$

$\sf \dfrac{ w^2 -7w +12 }{w^2 + w -2 0}$

So, the correct answer is:

$\sf B) \dfrac{ w^2 -7w +12 }{w^2 + w -2 0}$

Answer 2

Answer:

$\textsf{B)}\quad \dfrac{w^2-7w+12}{w^2+w-20}$

Step-by-step explanation:

To change the fraction (w - 3)/(w + 5) into an equivalent fraction with the denominator (w² + w - 20) we first need to factorize the new denominator and express it as a product of linear factors.

To factor the quadratic expression w² + w - 20, we need to find two numbers that multiply to the product of the leading coefficient and the constant term, and that sum to the coefficient of the middle term.

The product of the quadratic and constant terms is -20, and the coefficient of the middle term is 1. Therefore, the two numbers that multiply to -20 and sum to 1 are 5 and -4.

Factor the new denominator:

$\begin{aligned}w^2+w-20&=w^2+5w-4w-20\\&=w(w+5)-4(w+5)\\&=(w+5)(w-4)\end{aligned}$

Now, multiply the numerator and denominator of the original fraction by the missing factor in the new denominator, which is (w - 4). This gives us:

$\dfrac{w-3}{w+5}=\dfrac{(w-3)(w-4)}{(w+5)(w-4)}$

Finally, expand the numerator and denominator:

$\dfrac{(w-3)(w-4)}{(w+5)(w-4)}=\dfrac{w^2-7w+12}{x^2+w-20}$

Therefore, the equivalent fraction is:

$\Large\boxed{\boxed{\dfrac{w^2-7w+12}{x^2+w-20}}}$