Precalculus Examples

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

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Let $u = 2^{x}$ . Substitute $u$ for all occurrences of $2^{x}$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

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 1.4.2

Write the factored form using these integers.

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

Step 1.5

Replace all occurrences of $u$ with $2^{x}$ .

$(2^{x} - 2) (2^{x} + 6) = 0$

Step 2

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

$2^{x} - 2 = 0$

$2^{x} + 6 = 0$

Step 3

Set $2^{x} - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $2^{x} - 2$ equal to $0$ .

$2^{x} - 2 = 0$

Step 3.2

Solve $2^{x} - 2 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Add $2$ to both sides of the equation.

$2^{x} = 2$

Step 3.2.2

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

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

Step 3.2.3

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

$x = 1$

Step 4

Set $2^{x} + 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $2^{x} + 6$ equal to $0$ .

$2^{x} + 6 = 0$

Step 4.2

Solve $2^{x} + 6 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $6$ from both sides of the equation.

$2^{x} = - 6$

Step 4.2.2

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

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

Step 4.2.3

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

Undefined

Step 4.2.4

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

No solution

Step 5

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

$x = 1$