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(3)/(x^(2)-x-20)+(1)/(x^(2)+9x+20)
Which expression is equivalent to the sum for all
x > 5 ?
Choose 1 answer:
(A)
(4)/(2x^(2)+8x)
(B)
(4x+10)/((x^(2)-25)(x+4))
(c)
(4x+10)/((x-5)^(2)(x+4))
(D)
(4x+8)/((x^(2)-16)(x+5))

$\frac{3}{x^{2}-x-20}+\frac{1}{x^{2}+9 x+20}$ $\newline$ Which expression is equivalent to the sum for all x>5 ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{4}{2 x^{2}+8 x}$ $\newline$ (B) $\frac{4 x+10}{\left(x^{2}-25\right)(x+4)}$ $\newline$ (ᄃ) $\frac{4 x+10}{(x-5)^{2}(x+4)}$ $\newline$ (D) $\frac{4 x+8}{\left(x^{2}-16\right)(x+5)}$

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Compare linear and exponential growth

Full solution

Q.

\frac{3}{x^{2}-x-20}+\frac{1}{x^{2}+9 x+20}

\newline

Which expression is equivalent to the sum for all

x>5

?

\newline

Choose

1

answer:

\newline

(A)

\frac{4}{2 x^{2}+8 x}

\newline

(B)

\frac{4 x+10}{\left(x^{2}-25\right)(x+4)}

\newline

(ᄃ)

\frac{4 x+10}{(x-5)^{2}(x+4)}

\newline

(D)

\frac{4 x+8}{\left(x^{2}-16\right)(x+5)}

Factor Denominators: Factor the denominators. $\newline$ $x^2 - x - 20 = (x - 5)(x + 4)$ $\newline$ $x^2 + 9x + 20 = (x + 4)(x + 5)$
Rewrite Fractions: Rewrite the fractions with factored denominators. $\newline$ $\frac{3}{(x - 5)(x + 4)} + \frac{1}{(x + 4)(x + 5)}$
Find Common Denominator: Find a common denominator. $\newline$ $(x - 5)(x + 4)(x + 5)$
Rewrite with Common Denominator: Rewrite each fraction with the common denominator. $\newline$ $\frac{3(x + 5)}{(x - 5)(x + 4)(x + 5)} + \frac{1(x - 5)}{(x - 5)(x + 4)(x + 5)}$
Combine Fractions: Combine the fractions. $\newline$ $\frac{3(x + 5) + (x - 5)}{(x - 5)(x + 4)(x + 5)}$
Simplify Numerator: Simplify the numerator. $\newline$ $3(x + 5) + (x - 5) = 3x + 15 + x - 5 = 4x + 10$
Final Expression: Write the final expression. $\newline$ $\frac{4x + 10}{(x - 5)(x + 4)(x + 5)}$
Compare with Options: Compare with the given options. $\newline$ $(B) \frac{4x + 10}{(x^2 - 25)(x + 4)}$ $\newline$ Since $(x^2 - 25) = (x - 5)(x + 5)$ , option (B) matches.

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