Precalculus Examples

$3^{2 x} + 3^{x} - 30 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $3^{2 x}$ as ${(3^{x})}^{2}$ .

${(3^{x})}^{2} + 3^{x} - 30 = 0$

Step 1.2

Let $u = 3^{x}$ . Substitute $u$ for all occurrences of $3^{x}$ .

$u^{2} + u - 30 = 0$

Step 1.3

Factor $u^{2} + u - 30$ using the AC method.

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Step 1.3.1

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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 1.3.2

Write the factored form using these integers.

$(u - 5) (u + 6) = 0$

Step 1.4

Replace all occurrences of $u$ with $3^{x}$ .

$(3^{x} - 5) (3^{x} + 6) = 0$

Step 2

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

$3^{x} - 5 = 0$

$3^{x} + 6 = 0$

Step 3

Set $3^{x} - 5$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3^{x} - 5$ equal to $0$ .

$3^{x} - 5 = 0$

Step 3.2

Solve $3^{x} - 5 = 0$ for $x$ .

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Step 3.2.1

Add $5$ to both sides of the equation.

$3^{x} = 5$

Step 3.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{x}) = \ln (5)$

Step 3.2.3

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (5)$

Step 3.2.4

Divide each term in $x \ln (3) = \ln (5)$ by $\ln (3)$ and simplify.

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Step 3.2.4.1

Divide each term in $x \ln (3) = \ln (5)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (5)}{\ln (3)}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $\ln (3)$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (5)}{\ln (3)}$

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (5)}{\ln (3)}$

Step 4

Set $3^{x} + 6$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $3^{x} + 6$ equal to $0$ .

$3^{x} + 6 = 0$

Step 4.2

Solve $3^{x} + 6 = 0$ for $x$ .

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Step 4.2.1

Subtract $6$ from both sides of the equation.

$3^{x} = - 6$

Step 4.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{x}) = \ln (- 6)$

Step 4.2.3

The equation cannot be solved because $\ln (- 6)$ is undefined.

Undefined

Step 4.2.4

There is no solution for $3^{x} = - 6$

No solution

Step 5

The final solution is all the values that make $(3^{x} - 5) (3^{x} + 6) = 0$ true.

$x = \frac{\ln (5)}{\ln (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (5)}{\ln (3)}$

Decimal Form:

$x = 1.46497352 \dots$