(x^2-8x+15)^3x-1=(x^2-8x+15)^x+3

3 min read Jun 17, 2024
(x^2-8x+15)^3x-1=(x^2-8x+15)^x+3

Solving the Equation (x^2-8x+15)^3x-1=(x^2-8x+15)^x+3

This equation involves exponents with expressions as bases. To solve it, we'll utilize the following key principles:

1. Equating Exponents: If we have two powers with the same base and they are equal, their exponents must also be equal.

2. Factoring and Solving Equations: We can often simplify equations by factoring or rearranging terms.

Let's break down the solution step-by-step:

Step 1: Simplify the equation

Notice that both sides of the equation have the same base: (x^2 - 8x + 15). We can rewrite the equation to make it easier to work with:

(x^2 - 8x + 15)^(3x-1) = (x^2 - 8x + 15)^(x+3)

Step 2: Apply the principle of equating exponents

Since the bases are the same, we can equate the exponents:

3x - 1 = x + 3

Step 3: Solve for x

Now we have a simple linear equation. Let's solve for x:

  • 3x - x = 3 + 1
  • 2x = 4
  • x = 2

Step 4: Verify the Solution

It's always a good idea to check our solution by plugging it back into the original equation:

  • (2^2 - 8 * 2 + 15)^(3 * 2 - 1) = (2^2 - 8 * 2 + 15)^(2 + 3)
  • (4 - 16 + 15)^(5) = (4 - 16 + 15)^(5)
  • 3^5 = 3^5

The equation holds true, so our solution x = 2 is correct.

Therefore, the solution to the equation (x^2-8x+15)^3x-1=(x^2-8x+15)^x+3 is x = 2.

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