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Which of the following is equivalent to 2 x^2+8 x-24/2 x^2+20x-48
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16 Feb 2026, 13:53
16 Feb 2026, 13:53
Expert Reply
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Question Stats:
75% (01:51) correct
25% (01:01) wrong based on 12 sessions
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Which of the following is equivalent to \(\frac{2 x^2+8 x-24}{2 x^2+20 x-48}\) for all values of x for which both expressions are defined?
A. $\(\frac{x-2}{x-4}\)$
B. $\(\frac{x-2}{x+4}\)$
C. $\(\frac{x+2}{x+4}\)$
D. $\(\frac{x+6}{x-12}\)$
E. $\(\frac{x+6}{x+12}\)$
A. $\(\frac{x-2}{x-4}\)$
B. $\(\frac{x-2}{x+4}\)$
C. $\(\frac{x+2}{x+4}\)$
D. $\(\frac{x+6}{x-12}\)$
E. $\(\frac{x+6}{x+12}\)$
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Re: Which of the following is equivalent to 2 x^2+8 x-24/2 x^2+20x-48
[#permalink]
01 Mar 2026, 13:13
01 Mar 2026, 13:13
Expert Reply
Step-by-Step Solution:
1. Factor the Numerator $\(\left(2 x^2+8 x-24\right)\)$ :
- First, factor out the greatest common factor (GCF), which is 2 :
$$
\(2\left(x^2+4 x-12\right)\)
$$
- Now, factor the quadratic $\(x^2+4 x-12\)$. We need two numbers that multiply to -12 and add to 4 . Those numbers are +6 and -2 :
$$
\(2(x+6)(x-2)\)
$$
2. Factor the Denominator $\(\left(2 x^2+20 x-48\right)\)$ :
- First, factor out the GCF, which is 2:
$$
\(2\left(x^2+10 x-24\right)\)
$$
- Now, factor the quadratic $x^2+10 x-24$. We need two numbers that multiply to -24 and add to 10 . Those numbers are +12 and -2 :
$$
\(2(x+12)(x-2)\)
$$
3. Simplify the Expression:
Combine the factored forms:
$$
\(\frac{2(x+6)(x-2)}{2(x+12)(x-2)}\)
$$
- Cancel the 2 from the numerator and denominator.
- Cancel the common factor $\((x-2)\)$ (valid for all $\(x \neq 2\)$ ).
\(\begin{aligned}
&\text { The expression becomes: }\\
&\frac{x+6}{x+12}
\end{aligned}\)
1. Factor the Numerator $\(\left(2 x^2+8 x-24\right)\)$ :
- First, factor out the greatest common factor (GCF), which is 2 :
$$
\(2\left(x^2+4 x-12\right)\)
$$
- Now, factor the quadratic $\(x^2+4 x-12\)$. We need two numbers that multiply to -12 and add to 4 . Those numbers are +6 and -2 :
$$
\(2(x+6)(x-2)\)
$$
2. Factor the Denominator $\(\left(2 x^2+20 x-48\right)\)$ :
- First, factor out the GCF, which is 2:
$$
\(2\left(x^2+10 x-24\right)\)
$$
- Now, factor the quadratic $x^2+10 x-24$. We need two numbers that multiply to -24 and add to 10 . Those numbers are +12 and -2 :
$$
\(2(x+12)(x-2)\)
$$
3. Simplify the Expression:
Combine the factored forms:
$$
\(\frac{2(x+6)(x-2)}{2(x+12)(x-2)}\)
$$
- Cancel the 2 from the numerator and denominator.
- Cancel the common factor $\((x-2)\)$ (valid for all $\(x \neq 2\)$ ).
\(\begin{aligned}
&\text { The expression becomes: }\\
&\frac{x+6}{x+12}
\end{aligned}\)
gmatclubot
Re: Which of the following is equivalent to 2 x^2+8 x-24/2 x^2+20x-48 [#permalink]
01 Mar 2026, 13:13
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