Solve for x 5^(3x^2+12)=25^(-9x-6)

$5^{3 x^{2} + 12} = 25^{- 9 x - 6}$

Create equivalent expressions in the equation that all have equal bases.

$5^{3 x^{2} + 12} = 5^{2 (- 9 x - 6)}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 x^{2} + 12 = 2 (- 9 x - 6)$

Solve for $x$ .

Simplify $2 (- 9 x - 6)$ .

Rewrite.

$3 x^{2} + 12 = 0 + 0 + 2 (- 9 x - 6)$

Simplify by adding zeros.

$3 x^{2} + 12 = 2 (- 9 x - 6)$

Apply the distributive property.

$3 x^{2} + 12 = 2 (- 9 x) + 2 \cdot - 6$

Multiply.

Multiply $- 9$ by $2$ .

$3 x^{2} + 12 = - 18 x + 2 \cdot - 6$

Multiply $2$ by $- 6$ .

$3 x^{2} + 12 = - 18 x - 12$

Add $18 x$ to both sides of the equation.

$3 x^{2} + 12 + 18 x = - 12$

Add $12$ to both sides of the equation.

$3 x^{2} + 12 + 18 x + 12 = 0$

Add $12$ and $12$ .

$3 x^{2} + 18 x + 24 = 0$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} + 18 x + 24$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 18 x + 24 = 0$

Factor $3$ out of $18 x$ .

$3 (x^{2}) + 3 (6 x) + 24 = 0$

Factor $3$ out of $24$ .

$3 x^{2} + 3 (6 x) + 3 \cdot 8 = 0$

Factor $3$ out of $3 x^{2} + 3 (6 x)$ .

$3 (x^{2} + 6 x) + 3 \cdot 8 = 0$

Factor $3$ out of $3 (x^{2} + 6 x) + 3 \cdot 8$ .

$3 (x^{2} + 6 x + 8) = 0$

Factor.

Factor $x^{2} + 6 x + 8$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $6$ .

$2, 4$

Write the factored form using these integers.

$3 ((x + 2) (x + 4)) = 0$

Remove unnecessary parentheses.

$3 (x + 2) (x + 4) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x + 4 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $3 (x + 2) (x + 4) = 0$ true.

$x = - 2, - 4$

Solve for x 5^(3x^2+12)=25^(-9x-6)

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