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If
(x-5sqrta)(x+5sqrta) is equivalent to
x^(2)-375, what must be the value of
a ?

If $(x-5 \sqrt{a})(x+5 \sqrt{a})$ is equivalent to $x^{2}-375$ , what must be the value of $a$ ?

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Evaluate expression when a complex numbers and a variable term is given

Full solution

Q. If

(x-5 \sqrt{a})(x+5 \sqrt{a})

is equivalent to

x^{2}-375

, what must be the value of

a

?

Recognize as Difference of Squares: Given the expression $(x-5\sqrt{a})(x+5\sqrt{a})$ , we recognize it as a difference of squares, which takes the form $(A-B)(A+B) = A^2-B^2$ . Here, $A$ is $x$ and $B$ is $5\sqrt{a}$ .
Apply Formula: We apply the difference of squares formula to the given expression: $\newline$ $(x-5\sqrt{a})(x+5\sqrt{a}) = x^2 - (5\sqrt{a})^2$
Calculate Square: We calculate the square of $5\sqrt{a}$ : $\newline$ $(5\sqrt{a})^2 = (5)^2 \times (\sqrt{a})^2 = 25a$
Substitute Squared Term: We substitute the squared term back into the expression: $x^2 - (5\sqrt{a})^2 = x^2 - 25a$
Set Equal: According to the problem, this expression is equivalent to $x^2 - 375$ . Therefore, we set them equal to each other: $\newline$ $x^2 - 25a = x^2 - 375$
Cancel Out $x^2$ : Since $x^2$ appears on both sides of the equation, we can cancel it out, leaving us with: $-25a = -375$
Solve for $a$ : We solve for $a$ by dividing both sides by $-25$ : $a = \frac{-375}{-25}$
Final Value of a: Performing the division gives us the value of $a$ : $a = 15$

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