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(-1-5n+an^(2))-(7n-4n^(2)+9)=12n^(2)-12 n-10
If the given equation is true for all values of
n where
a is a constant, what is the value of
a ?
Choose 1 answer:
(A) 8
(B) 12
(C) 16
(D) 19

$\left(-1-5 n+a n^{2}\right)-\left(7 n-4 n^{2}+9\right)=12 n^{2}-12 n-10$ $\newline$ If the given equation is true for all values of $n$ where $a$ is a constant, what is the value of $a$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $8$ $\newline$ (B) $12$ $\newline$ (C) $16$ $\newline$ (D) $19$

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Full solution

Q.

\left(-1-5 n+a n^{2}\right)-\left(7 n-4 n^{2}+9\right)=12 n^{2}-12 n-10

\newline

If the given equation is true for all values of

n

where

a

is a constant, what is the value of

a

?

\newline

Choose

1

answer:

\newline

(A)

8

\newline

(B)

12

\newline

(C)

16

\newline

(D)

19

Distribute and Combine Terms: First, let's distribute the negative sign in the second set of parentheses and combine like terms on the left side of the equation. $\newline$ $(-1 - 5n + an^2) - (7n - 4n^2 + 9) = -1 - 5n + an^2 - 7n + 4n^2 - 9$
Combine Like Terms: Now, let's combine the like terms. $\newline$ $-1 - 5n - 7n + an^2 + 4n^2 - 9 = -10 - 12n + (a + 4)n^2$
Compare Coefficients: Next, we will compare the coefficients of the corresponding powers of $n$ on both sides of the equation. $\newline$ Left side: $-10 - 12n + (a + 4)n^2$ $\newline$ Right side: $-10 - 12n + 12n^2$
Set Coefficients Equal: For the equation to be true for all values of $n$ , the coefficients of the corresponding powers of $n$ must be equal on both sides. $\newline$ So, we set the coefficients of $n^2$ , $n$ , and the constant terms equal to each other. $\newline$ $(a + 4)n^2 = 12n^2$ $\newline$ $-12n = -12n$ $\newline$ $-10 = -10$
Solve for $a$ : Since the coefficients of $n$ and the constant terms are already equal, we only need to solve for $a$ from the equation involving $n^2$ terms. $\newline$ $a + 4 = 12$
Solve for $a$ : Since the coefficients of $n$ and the constant terms are already equal, we only need to solve for $a$ from the equation involving $n^2$ terms. $\newline$ $a + 4 = 12$ Subtract $4$ from both sides to solve for $a$ . $\newline$ $a = 12 - 4$ $\newline$ $a = 8$

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