Precalculus Examples

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

Step 1

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

$2^{2 x} + 2^{x} \cdot 2^{2} = 12$

Step 2

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

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

Step 3

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

$u^{2} + u \cdot 2^{2} = 12$

Step 4

Simplify each term.

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Step 4.1

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

$u^{2} + u \cdot 4 = 12$

Step 4.2

Move $4$ to the left of $u$ .

$u^{2} + 4 u = 12$

Step 5

Solve for $u$ .

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Step 5.1

Subtract $12$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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 $4$ .

$- 2, 6$

Step 5.2.2

Write the factored form using these integers.

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

Step 5.3

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

$u - 2 = 0$

$u + 6 = 0$

Step 5.4

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

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Step 5.4.1

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

$u - 2 = 0$

Step 5.4.2

Add $2$ to both sides of the equation.

$u = 2$

Step 5.5

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

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Step 5.5.1

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

$u + 6 = 0$

Step 5.5.2

Subtract $6$ from both sides of the equation.

$u = - 6$

Step 5.6

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

$u = 2, - 6$

Step 6

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

$2 = 2^{x}$

Step 7

Solve $2 = 2^{x}$ .

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Step 7.1

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

$2^{x} = 2$

Step 7.2

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

$2^{x} = 2^{1}$

Step 7.3

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

$x = 1$

Step 8

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

$- 6 = 2^{x}$

Step 9

Solve $- 6 = 2^{x}$ .

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Step 9.1

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

$2^{x} = - 6$

Step 9.2

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

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

Step 9.3

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

Undefined

Step 9.4

There is no solution for $2^{x} = - 6$

No solution

Step 10

List the solutions that makes the equation true.

$x = 1$