Solve for x 9^(x^2) = 3^(-5x-2)

-2,-1/2

x=-2,-1/2

change 9 to base 3 as 9 = 3^2

(3^2)^x^2 = 3^(-5x - 2)

so :
2x^2 = -5x-2
2x^2 + 5x + 2= 0
(2x + 1)(x + 2) = 0
x = -1/2 or x = -2

To solve for x in the equation 9^(x^2) = 3^(-5x-2), we will use the properties of exponents and logarithms. Follow these steps:

Step 1: Rewrite both sides of the equation using the base 3.
Since 9 is equal to 3^2, we can rewrite the left side of the equation as (3^2)^(x^2), which simplifies to 3^(2x^2).
Similarly, rewrite the right side of the equation as 3^(-5x-2).

Now our equation becomes: 3^(2x^2) = 3^(-5x-2).

Step 2: Equate the exponents.
Since the bases (3) on both sides are the same, the exponents must be equal. So we have:

2x^2 = -5x - 2.

Step 3: Rearrange the equation to get it in standard quadratic form.
We need to rewrite the equation in the form of ax^2 + bx + c = 0. So, rearrange the equation:

2x^2 + 5x + 2 = 0.

Step 4: Solve the quadratic equation.
We have a quadratic equation, so we can solve it using factoring, completing the square, or the quadratic formula. Let's solve it by factoring.

The equation factors as:

(2x + 1)(x + 2) = 0.

This equation will be true if either (2x + 1) = 0 or (x + 2) = 0. So we have two possibilities:

(2x + 1) = 0, which gives us x = -1/2.
(x + 2) = 0, which gives us x = -2.

Hence, the solutions to the equation 9^(x^2) = 3^(-5x-2) are x = -1/2 and x = -2.