For what values of [tex]$x$[/tex] does [tex]$25^x=5^{x^2-3}$[/tex]?
A. [tex]$x=-3, x=1$[/tex]
B. [tex]$x=-1, x=3$[/tex]
C. [tex]$x=-\frac{3}{2}, x-2$[/tex]
D. [tex]$x=-2, x=\frac{3}{2}$[/tex]

Rewrite the equation using the same base: 2 5 x = ( 5 2 ) x = 5 2 x .
Set the exponents equal to each other: 2 x = x 2 − 3 .
Rearrange to form a quadratic equation: x 2 − 2 x − 3 = 0 .
Solve the quadratic equation by factoring: ( x − 3 ) ( x + 1 ) = 0 , which gives the solutions x = − 1 and x = 3 . Thus, the answer is x = − 1 , x = 3 ​ .

Explanation

Problem Analysis We are given the equation 2 5 x = 5 x 2 − 3 and asked to find the values of x that satisfy it.

Rewriting the Equation Since 25 = 5 2 , we can rewrite the equation as ( 5 2 ) x = 5 x 2 − 3 .

Simplifying the Equation Using the power of a power rule, we simplify the left side to get 5 2 x = 5 x 2 − 3 .

Equating Exponents Since the bases are equal, the exponents must be equal, so 2 x = x 2 − 3 .

Forming a Quadratic Equation Rearranging the terms, we obtain a quadratic equation: x 2 − 2 x − 3 = 0 .

Factoring the Quadratic We can factor the quadratic equation as ( x − 3 ) ( x + 1 ) = 0 .

Finding the Solutions Thus, the solutions are x = 3 and x = − 1 .

Verification We can check these solutions in the original equation. If x = 3 , then 2 5 3 = ( 5 2 ) 3 = 5 6 and 5 3 2 − 3 = 5 9 − 3 = 5 6 , so x = 3 is a solution. If x = − 1 , then 2 5 − 1 = ( 5 2 ) − 1 = 5 − 2 and 5 ( − 1 ) 2 − 3 = 5 1 − 3 = 5 − 2 , so x = − 1 is a solution.

Final Answer Therefore, the values of x that satisfy the equation are x = − 1 and x = 3 .


Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to model compound interest. Suppose you invest an amount of money, P , at an annual interest rate, r , compounded annually. The amount of money you have after t years is given by A = P ( 1 + r ) t . This is an exponential equation, and solving for t would involve logarithms, which are closely related to exponential functions. Understanding how to solve exponential equations is crucial for making informed financial decisions.

Answered by GinnyAnswer | 2025-07-03

The values of x that satisfy the equation 2 5 x = 5 x 2 − 3 are x = − 1 and x = 3 . The correct multiple choice option is B. x = − 1 , x = 3 .
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Answered by Anonymous | 2025-07-04