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Which of the equations are true identities?
A.
(a+b)(2a+1)=a(2a+2b+1)
B.
(n+2)^(2)-n^(2)=4(n+1)
Choose 1 answer:
(A) Only A
(B) Only B
(C) Both A and B
(D) Neither A nor B

Which of the equations are true identities? $\newline$ A. $(a+b)(2 a+1)=a(2 a+2 b+1)$ $\newline$ B. $(n+2)^{2}-n^{2}=4(n+1)$ $\newline$ Choose $1$ answer: $\newline$ (A) Only A $\newline$ (B) Only B $\newline$ (C) Both A and B $\newline$ (D) Neither A nor B

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Full solution

Q. Which of the equations are true identities?

\newline

A.

(a+b)(2 a+1)=a(2 a+2 b+1)

\newline

B.

(n+2)^{2}-n^{2}=4(n+1)

\newline

Choose

1

answer:

\newline

(A) Only A

\newline

(B) Only B

\newline

(C) Both A and B

\newline

(D) Neither A nor B

Expand Expression: Expand $(a+b)(2a+1)$ using the distributive property. $\newline$ $(a+b)(2a+1) = a(2a) + a(1) + b(2a) + b(1)$ $\newline$ $= 2a^2 + a + 2ab + b$
Compare Expanded Forms: Compare the expanded form with the right side of equation A. $\newline$ $2a^2 + a + 2ab + b$ ?= $a(2a+2b+1)$ $\newline$ = $2a^2 + 2ab + a$
Identify Missing Term: Notice that the term ' extit{b}' is missing on the right side of equation extit{A}. Therefore, equation extit{A} is not a true identity.
Apply Binomial Theorem: Expand $(n+2)^2$ using the binomial theorem. $\newline$ $(n+2)^2 = n^2 + 2\cdot 2n + 2^2$ $\newline$ $= n^2 + 4n + 4$
Subtract and Simplify: Subtract $n^2$ from both sides of equation B. $\newline$ $(n+2)^2 - n^2 = 4n + 4$
Compare Results: Compare the result with the right side of equation B. $\newline$ $4n + 4 \neq 4(n+1)$ $\newline$ $= 4n + 4$
Verify Identity: Since both sides of equation $B$ are equal after simplification, equation $B$ is a true identity.

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