$(a x+5)(x+v)=a x^{2}+25 x+25$ $\newline$ What is the value of $a$ in the given equation? $\newline$ Choose $1$ answer: $\newline$ (A) $-5$ $\newline$ (B) $-4$ $\newline$ (C) $4$ $\newline$ (D) $5$

Full solution

(a x+5)(x+v)=a x^{2}+25 x+25

\newline

What is the value of

a

in the given equation?

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Choose

1

answer:

\newline

(A)

-5

\newline

(B)

-4

\newline

(C)

4

\newline

(D)

5

Expand Left Side: Expand the left side of the equation $(ax+5)(x+v)$ using the distributive property (FOIL method). $(ax+5)(x+v) = ax\cdot x + ax\cdot v + 5\cdot x + 5\cdot v$
Compare with Right Side: Compare the expanded form with the right side of the equation $ax^2 + 25x + 25$ . We have $ax \times x = ax^2$ , which is already in the correct form. Now, we need to find the terms that combine to give us $25x$ on the right side.
Find Terms for $25x$ : The terms $ax*v$ and $5*x$ must add up to $25x$ . $\newline$ So, $ax*v + 5*x = 25x$ .
Deduce Value of $v$ : Since there is no $x$ term on the left side that would correspond to $25x$ on the right side, we can deduce that $v$ must be $5$ to satisfy the equation. $\newline$ This is because $5\times x$ is already part of the $25x$ term on the right side, so $ax\times v$ must be the remaining $20x$ .
Simplify Equation: Now we have $ax \times 5 + 5 \times x = 20x + 5x = 25x$ . This simplifies to $5a \times x + 5 \times x = 25x$ .
Divide and Solve for $a$ : Divide the entire equation by $x$ to simplify and solve for $a$ . $5a + 5 = 25$ .
Subtract to Isolate $a$ : Subtract $5$ from both sides to isolate the term with $a$ . $5a = 25 - 5.$
Calculate Value of a: Calculate the value of a. $\newline$ $5a = 20$ . $\newline$ $a = \frac{20}{5}$ . $\newline$ $a = 4$ .
Final Answer: The value of $a$ is $4$ , which corresponds to option $(C)$ .