If m^(2)+p^(2)=x and 4mp=y, which of the following is equivalent to 4x+2y ? Choose 1 answer: (A) (2m+p)^(2) (B) (2m+2p)^(2) (C) (4m+2p)^(2) (D) (4m+8p)^(2)

Given Equations: We are given two equations: 1. m^2 + p^2 = x 2. 4mp = y We need to find an equivalent expression for 4x + 2y using these equations.Express 4x: First, let's express 4x in terms of m and p using the first equation: 4x = 4(m^2 + p^2)Express 2y: Now, let's express 2y in terms of m and p using the second equation: 2y = 2(4mp)Combine Expressions: Combine the expressions for 4x and 2y: 4x + 2y = 4(m^2 + p^2) + 2(4mp)Simplify Expression: Simplify the expression by distributing the constants: 4x + 2y = 4m^2 + 4p^2 + 8mpFactor Expression: Now, let's try to factor this expression to match one of the answer choices. We are looking for a perfect square since all the answer choices are in the form of a squared binomial.Identify Perfect Square: We notice that 4m^2 + 4p^2 + 8mp can be written as (2m + 2p)^2 because: (2m + 2p)^2 = (2m)^2 + 2*(2m)*(2p) + (2p)^2 = 4m^2 + 8mp + 4p^2 Which matches our expression for 4x + 2y.Final Equivalent Expression: Therefore, the equivalent expression for 4x + 2y is (2m + 2p)^2, which corresponds to answer choice (B).

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If $m^{2}+p^{2}=x$ and $4mp=y$ , which of the following is equivalent to $4x+2y$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $(2m+p)^{2}$ $\newline$ (B) $(2m+2p)^{2}$ $\newline$ (C) $(4m+2p)^{2}$ $\newline$ (D) $(4m+8p)^{2}$

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Algebra 1

Transformations of functions

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Q. If

m^{2}+p^{2}=x

and

4mp=y

, which of the following is equivalent to

4x+2y

\newline

Choose

1

answer:

\newline

(A)

(2m+p)^{2}

\newline

(B)

(2m+2p)^{2}

\newline

(C)

(4m+2p)^{2}

\newline

(D)

(4m+8p)^{2}

Given Equations: We are given two equations: $\newline$ $1$ . $m^2 + p^2 = x$ $\newline$ $2$ . $4mp = y$ $\newline$ We need to find an equivalent expression for $4x + 2y$ using these equations.
Express $4x$ : First, let's express $4x$ in terms of $m$ and $p$ using the first equation: $\newline$ $4x = 4(m^2 + p^2)$
Express $2y$ : Now, let's express $2y$ in terms of $m$ and $p$ using the second equation: $\newline$ $2y = 2(4mp)$
Combine Expressions: Combine the expressions for $4x$ and $2y$ : $\newline$ $4x + 2y = 4(m^2 + p^2) + 2(4mp)$
Simplify Expression: Simplify the expression by distributing the constants: $4x + 2y = 4m^2 + 4p^2 + 8mp$
Factor Expression: Now, let's try to factor this expression to match one of the answer choices. We are looking for a perfect square since all the answer choices are in the form of a squared binomial.
Identify Perfect Square: We notice that $4m^2 + 4p^2 + 8mp$ can be written as $(2m + 2p)^2$ because: $\newline$ ($2$m + $2$p)^$2$ = ($2$m)^$2$ + $2$\times($2$m)\times($2$p) + ($2$p)^$2$$\newline$ = $4$m^$2$ + $8$mp + $4$p^$2$ $\newline$ Which matches our expression for $4x + 2y$ .
Final Equivalent Expression: Therefore, the equivalent expression for $4x + 2y$ is $(2m + 2p)^2$ , which corresponds to answer choice $(B)$ .