((5)/(3))^(x+1)*((9)/(25))^(x^(2)+2x-11)=((5)/(3))^(9)

2 min read Jun 06, 2024

Solving the Exponential Equation

In this article, we will solve the exponential equation:

$\left(\frac{5}{3}\right)^{x+1} \times \left(\frac{9}{25}\right)^{x^2 + 2x - 11} = \left(\frac{5}{3}\right)^9$

First, we can simplify the equation by combining the exponential terms:

$\left(\frac{5}{3}\right)^{x+1} \times \left(\frac{3}{5}\right)^{2(x^2 + 2x - 11)} = \left(\frac{5}{3}\right)^9$

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

$x+1 = 9$ $2(x^2 + 2x - 11) = 9$

Now, we can solve the quadratic equation:

$2(x^2 + 2x - 11) = 9$ $2x^2 + 4x - 44 = 9$ $2x^2 + 4x - 53 = 0$

Using the quadratic formula, we get:

$x = \frac{-4 \pm \sqrt{16 - 4(2)(-53)}}{4}$ $x = \frac{-4 \pm \sqrt{424}}{4}$ $x = \frac{-4 \pm 2\sqrt{106}}{4}$ $x = -1 \pm \sqrt{\frac{106}{4}}$

Therefore, we have found the solution to the exponential equation:

$x = -1 \pm \sqrt{\frac{106}{4}}$

This is the final solution to the equation.