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$(3y-2)(y+a)=3y^{2}+b$ ? $\newline$ If the given equation is true for all values of $y$ , where $a$ and $b$ are constants, which of the following is the value of $b$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $-38$ $\newline$ (B) $12$ $\newline$ (C) $34$ $\newline$ (D) $36$

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Full solution

Q.

(3y-2)(y+a)=3y^{2}+b

?

\newline

If the given equation is true for all values of

y

, where

a

and

b

are constants, which of the following is the value of

b

?

\newline

Choose

1

answer:

\newline

(A)

-38

\newline

(B)

12

\newline

(C)

34

\newline

(D)

36

Expand and Compare: We need to expand the left side of the equation $(3y-2)(y+a)$ to compare it with the right side $3y^2+b$ . $(3y-2)(y+a) = 3y(y) + 3y(a) - 2(y) - 2(a)$
Simplify Expression: Now we simplify the expression by multiplying the terms. $\newline$ $3y(y) + 3y(a) - 2(y) - 2(a) = 3y^2 + 3ay - 2y - 2a$
Compare with Right Side: We compare the simplified expression with the right side of the equation. $\newline$ $3y^2 + 3ay - 2y - 2a = 3y^2 + b$
Constant Term Comparison: Since the equation is true for all values of $y$ , the coefficients of the corresponding powers of $y$ on both sides must be equal. This means that the constant term on the left side, $-2a$ , must be equal to the constant term $b$ on the right side. $\newline$ $-2a = b$
Conclusion: We do not have the value of $a$ , but we are not asked to find $a$ . We are asked to find $b$ , which is the constant term in the equation. Since we cannot determine the exact value of $b$ without knowing $a$ , we cannot proceed further in solving for $b$ . However, we can conclude that $b$ is equal to $-2a$ , whatever the value of $a$ might be.

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