Question Video: Solving Exponential Equations Involving Quadratic Expressions Using Laws of Exponents Mathematics

Find the solution set 3^(𝑥² − 36) = 9^(𝑥² − 36) in ℝ.

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Video Transcript

Find the solution set of three to the power of 𝑥 squared minus 36 is equal to nine to the power of 𝑥 squared minus 36 for all real numbers.

We know that three squared is equal to nine. This means that we can rewrite the right-hand side of our equation as three squared to the power of 𝑥 squared minus 36, which is equal to three to the power of two multiplied by 𝑥 squared minus 36. At this stage, we have three raised to a power on both sides of our equation. This means that the exponents must be equal. 𝑥 squared minus 36 is equal to two multiplied by 𝑥 squared minus 36.

On the right-hand side, we can distribute the parentheses. We can then add 72 and subtract 𝑥 squared from both sides of this equation. This gives us 𝑥 squared is equal to 36. We can then square root both sides, recalling that we can have a positive or negative answer. The square root of 36 equals six. Therefore, 𝑥 is equal to positive or negative six. The solution set of three to the power of 𝑥 squared minus 36 is equal to nine to the power of 𝑥 squared minus 36 contains the values negative six and six.

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