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

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

Step 1

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

$3^{x^{2} + 5 x} = 3^{2 \cdot - 3}$

Step 2

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

$x^{2} + 5 x = 2 \cdot - 3$

Step 3

Solve for $x$ .

Multiply $2$ by $- 3$ .

$x^{2} + 5 x = - 6$

Add $6$ to both sides of the equation.

$x^{2} + 5 x + 6 = 0$

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

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

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

$x + 2 = 0$

$x + 3 = 0$

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 $x + 3$ 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 $(x + 2) (x + 3) = 0$ true.

$x = - 2, - 3$

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

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