Solve for x 9^(-3x-6)=3^(3x^2-21)

$9^{- 3 x - 6} = 3^{3 x^{2} - 21}$

Step 1

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

$3^{2 (- 3 x - 6)} = 3^{3 x^{2} - 21}$

Step 2

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

$2 (- 3 x - 6) = 3 x^{2} - 21$

Step 3

Solve for $x$ .

Simplify $2 (- 3 x - 6)$ .

Rewrite.

$0 + 0 + 2 (- 3 x - 6) = 3 x^{2} - 21$

Simplify by adding zeros.

$2 (- 3 x - 6) = 3 x^{2} - 21$

Apply the distributive property.

$2 (- 3 x) + 2 \cdot - 6 = 3 x^{2} - 21$

Multiply.

Multiply $- 3$ by $2$ .

$- 6 x + 2 \cdot - 6 = 3 x^{2} - 21$

Multiply $2$ by $- 6$ .

$- 6 x - 12 = 3 x^{2} - 21$

Subtract $3 x^{2}$ from both sides of the equation.

$- 6 x - 12 - 3 x^{2} = - 21$

Add $21$ to both sides of the equation.

$- 6 x - 12 - 3 x^{2} + 21 = 0$

Add $- 12$ and $21$ .

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

Factor the left side of the equation.

Factor $- 3$ out of $- 6 x - 3 x^{2} + 9$ .

Reorder $- 6 x$ and $- 3 x^{2}$ .

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

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

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

Factor $- 3$ out of $- 6 x$ .

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

Factor $- 3$ out of $9$ .

$- 3 x^{2} - 3 (2 x) - 3 \cdot - 3 = 0$

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

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

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

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

Factor.

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Write the factored form using these integers.

$- 3 ((x - 1) (x + 3)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 1) (x + 3) = 0$

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

$x - 1 = 0$

$x + 3 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $- 3 (x - 1) (x + 3) = 0$ true.

$x = 1, - 3$

Solve for x 9^(-3x-6)=3^(3x^2-21)

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