Calculus Examples

$3^{2 x} - 3^{x + 2} + 8 = 0$

Step 1

Rewrite $3^{x + 2}$ as $3^{x} \cdot 3^{2}$ .

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

Step 2

Rewrite $3^{2 x}$ as exponentiation.

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

Step 3

Remove parentheses.

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

Step 4

Substitute $u$ for $3^{x}$ .

$u^{2} - u \cdot 3^{2} + 8 = 0$

Step 5

Simplify each term.

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Step 5.1

Raise $3$ to the power of $2$ .

$u^{2} - u \cdot 9 + 8 = 0$

Step 5.2

Multiply $9$ by $- 1$ .

$u^{2} - 9 u + 8 = 0$

Step 6

Solve for $u$ .

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Step 6.1

Factor $u^{2} - 9 u + 8$ using the AC method.

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Step 6.1.1

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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Step 6.1.2

Write the factored form using these integers.

$(u - 8) (u - 1) = 0$

Step 6.2

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

$u - 8 = 0$

$u - 1 = 0$

Step 6.3

Set $u - 8$ equal to $0$ and solve for $u$ .

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Step 6.3.1

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

Step 6.3.2

Add $8$ to both sides of the equation.

$u = 8$

Step 6.4

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

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Step 6.4.1

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

$u - 1 = 0$

Step 6.4.2

Add $1$ to both sides of the equation.

$u = 1$

Step 6.5

The final solution is all the values that make $(u - 8) (u - 1) = 0$ true.

$u = 8, 1$

Step 7

Substitute $8$ for $u$ in $u = 3^{x}$ .

$8 = 3^{x}$

Step 8

Solve $8 = 3^{x}$ .

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Step 8.1

Rewrite the equation as $3^{x} = 8$ .

$3^{x} = 8$

Step 8.2

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

$\ln (3^{x}) = \ln (8)$

Step 8.3

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

$x \ln (3) = \ln (8)$

Step 8.4

Divide each term in $x \ln (3) = \ln (8)$ by $\ln (3)$ and simplify.

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Step 8.4.1

Divide each term in $x \ln (3) = \ln (8)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (8)}{\ln (3)}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

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

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (8)}{\ln (3)}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (8)}{\ln (3)}$

Step 9

Substitute $1$ for $u$ in $u = 3^{x}$ .

$1 = 3^{x}$

Step 10

Solve $1 = 3^{x}$ .

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Step 10.1

Rewrite the equation as $3^{x} = 1$ .

$3^{x} = 1$

Step 10.2

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

$\ln (3^{x}) = \ln (1)$

Step 10.3

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

$x \ln (3) = \ln (1)$

Step 10.4

Simplify the right side.

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Step 10.4.1

The natural logarithm of $1$ is $0$ .

$x \ln (3) = 0$

Step 10.5

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ and simplify.

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Step 10.5.1

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 10.5.2

Simplify the left side.

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Step 10.5.2.1

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

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Step 10.5.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Step 10.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{\ln (3)}$

Step 10.5.3

Simplify the right side.

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Step 10.5.3.1

Divide $0$ by $\ln (3)$ .

$x = 0$

Step 11

List the solutions that makes the equation true.

$x = \frac{\ln (8)}{\ln (3)}, 0$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (8)}{\ln (3)}, 0$

Decimal Form:

$x = 1.89278926 \dots, 0$