Solve for x 9^(-3x-84)=3^(-3x^2-9x)

$9^{- 3 x - 84} = 3^{- 3 x^{2} - 9 x}$

Step 1

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

$3^{2 (- 3 x - 84)} = 3^{- 3 x^{2} - 9 x}$

Step 2

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

$2 (- 3 x - 84) = - 3 x^{2} - 9 x$

Step 3

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 3 x^{2} - 9 x = 2 (- 3 x - 84)$

Simplify $2 (- 3 x - 84)$ .

Apply the distributive property.

$- 3 x^{2} - 9 x = 2 (- 3 x) + 2 \cdot - 84$

Multiply.

Multiply $- 3$ by $2$ .

$- 3 x^{2} - 9 x = - 6 x + 2 \cdot - 84$

Multiply $2$ by $- 84$ .

$- 3 x^{2} - 9 x = - 6 x - 168$

Move all terms containing $x$ to the left side of the equation.

Add $6 x$ to both sides of the equation.

$- 3 x^{2} - 9 x + 6 x = - 168$

Add $- 9 x$ and $6 x$ .

$- 3 x^{2} - 3 x = - 168$

Add $168$ to both sides of the equation.

$- 3 x^{2} - 3 x + 168 = 0$

Factor the left side of the equation.

Factor $- 3$ out of $- 3 x^{2} - 3 x + 168$ .

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

$- 3 x^{2} - 3 x + 168 = 0$

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

$- 3 x^{2} - 3 x + 168 = 0$

Factor $- 3$ out of $168$ .

$- 3 x^{2} - 3 x - 3 \cdot - 56 = 0$

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

$- 3 (x^{2} + x) - 3 \cdot - 56 = 0$

Factor $- 3$ out of $- 3 (x^{2} + x) - 3 (- 56)$ .

$- 3 (x^{2} + x - 56) = 0$

Factor.

Factor $x^{2} + x - 56$ 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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Write the factored form using these integers.

$- 3 ((x - 7) (x + 8)) = 0$

Remove unnecessary parentheses.

$- 3 (x - 7) (x + 8) = 0$

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

$x - 7 = 0$

$x + 8 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $- 3 (x - 7) (x + 8) = 0$ true.

$x = 7, - 8$

Solve for x 9^(-3x-84)=3^(-3x^2-9x)

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