Jasonbradley
Home

(1/7)^x^2-2x-3=7^x+1

3 min read Jun 16, 2024
(1/7)^x^2-2x-3=7^x+1

Solving the Exponential Equation: (1/7)^x^2-2x-3 = 7^x+1

This article will guide you through the process of solving the exponential equation:

(1/7)^x^2-2x-3 = 7^x+1

Understanding the Basics

Before we jump into the solution, let's recall some important properties of exponents:

  • Negative exponents: a^-n = 1/a^n
  • Fractional exponents: a^(m/n) = (√n a)^m
  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n)

Simplifying the Equation

  1. Rewrite (1/7) as 7^-1: The equation becomes: (7^-1)^x^2-2x-3 = 7^x+1

  2. Apply the power of a power rule: (a^m)^n = a^(m*n) 7^(-x^2+2x+3) = 7^x+1

  3. Equate the exponents: Since the bases are the same, we can equate the exponents. -x^2 + 2x + 3 = x + 1

Solving the Quadratic Equation

  1. Move all terms to one side: -x^2 + x + 2 = 0

  2. Solve the quadratic equation: This can be solved using factoring, the quadratic formula, or completing the square. Factoring yields: (-x + 2)(x + 1) = 0 Therefore, x = 2 or x = -1

Verifying the Solutions

It's crucial to verify our solutions by plugging them back into the original equation.

  1. For x = 2: (1/7)^2^2-2*2-3 = 7^2+1 (1/7)^-3 = 7^3 7^3 = 7^3 (This solution is valid)

  2. For x = -1: (1/7)^(-1)^2-2(-1)-3 = 7^(-1)+1* (1/7)^0 = 7^0 1 = 1 (This solution is also valid)

Conclusion

Therefore, the solutions to the equation (1/7)^x^2-2x-3 = 7^x+1 are x = 2 and x = -1.

Featured Posts