Finite Math Examples

$2^{2 x} - 2^{x + 2} - 12 = 0$

Step 1

Reorder terms.

$2^{2 x} - 12 - 2^{x + 2} = 0$

Step 2

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

$2^{2 x} - 12 - (2^{x} \cdot 2^{2}) = 0$

Step 3

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

${(2^{x})}^{2} - 12 - (2^{x} \cdot 2^{2}) = 0$

Step 4

Remove parentheses.

${(2^{x})}^{2} - 12 - 2^{x} \cdot 2^{2} = 0$

Step 5

Substitute $u$ for $2^{x}$ .

$u^{2} - 12 - u \cdot 2^{2} = 0$

Step 6

Simplify each term.

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Step 6.1

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

$u^{2} - 12 - u \cdot 4 = 0$

Step 6.2

Multiply $4$ by $- 1$ .

$u^{2} - 12 - 4 u = 0$

Step 7

Move $- 12$ .

$u^{2} - 4 u - 12 = 0$

Step 8

Solve for $u$ .

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Step 8.1

Factor $u^{2} - 4 u - 12$ using the AC method.

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Step 8.1.1

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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 8.1.2

Write the factored form using these integers.

$(u - 6) (u + 2) = 0$

Step 8.2

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

$u - 6 = 0$

$u + 2 = 0$

Step 8.3

Set $u - 6$ equal to $0$ and solve for $u$ .

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Step 8.3.1

Set $u - 6$ equal to $0$ .

$u - 6 = 0$

Step 8.3.2

Add $6$ to both sides of the equation.

$u = 6$

Step 8.4

Set $u + 2$ equal to $0$ and solve for $u$ .

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Step 8.4.1

Set $u + 2$ equal to $0$ .

$u + 2 = 0$

Step 8.4.2

Subtract $2$ from both sides of the equation.

$u = - 2$

Step 8.5

The final solution is all the values that make $(u - 6) (u + 2) = 0$ true.

$u = 6, - 2$

Step 9

Substitute $6$ for $u$ in $u = 2^{x}$ .

$6 = 2^{x}$

Step 10

Solve $6 = 2^{x}$ .

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Step 10.1

Rewrite the equation as $2^{x} = 6$ .

$2^{x} = 6$

Step 10.2

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

$\ln (2^{x}) = \ln (6)$

Step 10.3

Expand $\ln (2^{x})$ by moving $x$ outside the logarithm.

$x \ln (2) = \ln (6)$

Step 10.4

Divide each term in $x \ln (2) = \ln (6)$ by $\ln (2)$ and simplify.

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Step 10.4.1

Divide each term in $x \ln (2) = \ln (6)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (6)}{\ln (2)}$

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

Cancel the common factor of $\ln (2)$ .

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Step 10.4.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (6)}{\ln (2)}$

Step 10.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{\ln (2)}$

Step 11

Substitute $- 2$ for $u$ in $u = 2^{x}$ .

$- 2 = 2^{x}$

Step 12

Solve $- 2 = 2^{x}$ .

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Step 12.1

Rewrite the equation as $2^{x} = - 2$ .

$2^{x} = - 2$

Step 12.2

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

$\ln (2^{x}) = \ln (- 2)$

Step 12.3

The equation cannot be solved because $\ln (- 2)$ is undefined.

Undefined

Step 12.4

There is no solution for $2^{x} = - 2$

No solution

Step 13

List the solutions that makes the equation true.

$x = \frac{\ln (6)}{\ln (2)}$