Precalculus Examples

$4^{2 x + 1} - 17 \cdot 4^{x} + 4 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Rewrite $4^{2 x + 1}$ as $4^{2 x} \cdot 4^{1}$ .

$4^{2 x} \cdot 4 - 17 \cdot 4^{x} + 4 = 0$

Step 1.2

Rewrite $4^{2 x}$ as ${(4^{x})}^{2}$ .

${(4^{x})}^{2} \cdot 4 - 17 \cdot 4^{x} + 4 = 0$

Step 1.3

Let $u = 4^{x}$ . Substitute $u$ for all occurrences of $4^{x}$ .

Tap for more steps...

Step 1.3.1

Evaluate the exponent.

$u^{2} \cdot 4 - 17 \cdot u + 4 = 0$

Step 1.3.2

Move $4$ to the left of $u^{2}$ .

$4 u^{2} - 17 u + 4 = 0$

Step 1.4

Factor by grouping.

Tap for more steps...

Step 1.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 4 = 16$ and whose sum is $b = - 17$ .

Tap for more steps...

Step 1.4.1.1

Factor $- 17$ out of $- 17 u$ .

$4 u^{2} - 17 u + 4 = 0$

Step 1.4.1.2

Rewrite $- 17$ as $- 1$ plus $- 16$

$4 u^{2} + (- 1 - 16) u + 4 = 0$

Step 1.4.1.3

Apply the distributive property.

$4 u^{2} - 1 u - 16 u + 4 = 0$

Step 1.4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.4.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) - 16 u + 4 = 0$

Step 1.4.2.2

Factor out the greatest common factor (GCF) from each group.

$u (4 u - 1) - 4 (4 u - 1) = 0$

Step 1.4.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (u - 4) = 0$

Step 1.5

Replace all occurrences of $u$ with $4^{x}$ .

$(4 \cdot 4^{x} - 1) (4^{x} - 4) = 0$

Step 2

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

$4 \cdot 4^{x} - 1 = 0$

$4^{x} - 4 = 0$

Step 3

Set $4 \cdot 4^{x} - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $4 \cdot 4^{x} - 1$ equal to $0$ .

$4 \cdot 4^{x} - 1 = 0$

Step 3.2

Solve $4 \cdot 4^{x} - 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Multiply $4$ by $4^{x}$ .

Tap for more steps...

Step 3.2.1.1

Raise $4$ to the power of $1$ .

$4^{1} \cdot 4^{x} - 1 = 0$

Step 3.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4^{1 + x} - 1 = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$4^{1 + x} = 1$

Step 3.2.3

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

$\ln (4^{1 + x}) = \ln (1)$

Step 3.2.4

Expand $\ln (4^{1 + x})$ by moving $1 + x$ outside the logarithm.

$(1 + x) \ln (4) = \ln (1)$

Step 3.2.5

Simplify the left side.

Tap for more steps...

Step 3.2.5.1

Simplify $(1 + x) \ln (4)$ .

Tap for more steps...

Step 3.2.5.1.1

Apply the distributive property.

$1 \ln (4) + x \ln (4) = \ln (1)$

Step 3.2.5.1.2

Multiply $\ln (4)$ by $1$ .

$\ln (4) + x \ln (4) = \ln (1)$

Step 3.2.6

Simplify the right side.

Tap for more steps...

Step 3.2.6.1

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

$\ln (4) + x \ln (4) = 0$

Step 3.2.7

Reorder $\ln (4)$ and $x \ln (4)$ .

$x \ln (4) + \ln (4) = 0$

Step 3.2.8

Subtract $\ln (4)$ from both sides of the equation.

$x \ln (4) = - \ln (4)$

Step 3.2.9

Divide each term in $x \ln (4) = - \ln (4)$ by $\ln (4)$ and simplify.

Tap for more steps...

Step 3.2.9.1

Divide each term in $x \ln (4) = - \ln (4)$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (4)}{\ln (4)}$

Step 3.2.9.2

Simplify the left side.

Tap for more steps...

Step 3.2.9.2.1

Cancel the common factor of $\ln (4)$ .

Tap for more steps...

Step 3.2.9.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (4)}{\ln (4)}$

Step 3.2.9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (4)}{\ln (4)}$

Step 3.2.9.3

Simplify the right side.

Tap for more steps...

Step 3.2.9.3.1

Cancel the common factor of $\ln (4)$ .

Tap for more steps...

Step 3.2.9.3.1.1

Cancel the common factor.

$x = \frac{- \ln (4)}{\ln (4)}$

Step 3.2.9.3.1.2

Divide $- 1$ by $1$ .

$x = - 1$

Step 4

Set $4^{x} - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $4^{x} - 4$ equal to $0$ .

$4^{x} - 4 = 0$

Step 4.2

Solve $4^{x} - 4 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Add $4$ to both sides of the equation.

$4^{x} = 4$

Step 4.2.2

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

$4^{x} = 4^{1}$

Step 4.2.3

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

$x = 1$

Step 5

The final solution is all the values that make $(4 \cdot 4^{x} - 1) (4^{x} - 4) = 0$ true.

$x = - 1, 1$