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(x-3)(ax+4)=0
In the given equation,
a is a constant. If the equation has the solutions
x=3 and
x=-2, what is the value of
a ?

$(x-3)(a x+4)=0$ $\newline$ In the given equation, $a$ is a constant. If the equation has the solutions $x=3$ and $x=-2$ , what is the value of $a$ ?

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Write and solve direct variation equations

Full solution

Q.

(x-3)(a x+4)=0

\newline

In the given equation,

a

is a constant. If the equation has the solutions

x=3

and

x=-2

, what is the value of

a

?

Identify Solution for $x=3$ : We are given that $(x-3)(ax+4)=0$ has solutions $x=3$ and $x=-2$ . We will first use the fact that $x=3$ is a solution to find a relationship between $a$ and the constants in the equation. $\newline$ Substitute $x=3$ into the equation: $\newline$ $(3-3)(a\cdot 3+4)=0$ $\newline$ $0\cdot(3a+4)=0$ $\newline$ Since $0$ multiplied by any number is $0$ , this does not provide information about $a$ . This step confirms that $x=3$ is indeed a solution.
Utilize Solution for $x=-2$ : Next, we will use the fact that $x=-2$ is also a solution to the equation. Substitute $x=-2$ into the equation: $\newline$ $(-2-3)(a*(-2)+4)=0$ $\newline$ $(-5)(-2a+4)=0$ $\newline$ Since the product is zero, one of the factors must be zero. Therefore, we set the second factor equal to zero to find $a$ : $\newline$ $-2a+4=0$
Solve for a: Now, solve for a: $\newline$ Add $2a$ to both sides: $\newline$ $4=2a$ $\newline$ Divide both sides by $2$ : $\newline$ $a=2$

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