Precalculus Examples

$4^{2 x} - 5 \cdot 4^{x + \frac{1}{2}} + 16 = 0$

Step 1

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

$4^{2 x} - 5 \cdot (4^{x} \cdot 4^{\frac{1}{2}}) + 16 = 0$

Step 2

Rewrite $4^{2 x}$ as exponentiation.

${(4^{x})}^{2} - 5 \cdot (4^{x} \cdot 4^{\frac{1}{2}}) + 16 = 0$

Step 3

Remove parentheses.

${(4^{x})}^{2} - 5 \cdot 4^{x} \cdot 4^{\frac{1}{2}} + 16 = 0$

Step 4

Substitute $u$ for $4^{x}$ .

$u^{2} - 5 \cdot u \cdot 4^{\frac{1}{2}} + 16 = 0$

Step 5

Simplify each term.

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Step 5.1

Rewrite $4$ as $2^{2}$ .

$u^{2} - 5 u \cdot {(2^{2})}^{\frac{1}{2}} + 16 = 0$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{2} - 5 u \cdot 2^{2 (\frac{1}{2})} + 16 = 0$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$u^{2} - 5 u \cdot 2^{2 (\frac{1}{2})} + 16 = 0$

Step 5.3.2

Rewrite the expression.

$u^{2} - 5 u \cdot 2 + 16 = 0$

Step 5.4

Evaluate the exponent.

$u^{2} - 5 u \cdot 2 + 16 = 0$

Step 5.5

Multiply $2$ by $- 5$ .

$u^{2} - 10 u + 16 = 0$

Step 6

Solve for $u$ .

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Step 6.1

Factor $u^{2} - 10 u + 16$ 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 $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 6.1.2

Write the factored form using these integers.

$(u - 8) (u - 2) = 0$

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 $0$ .

$u - 8 = 0$

$u - 2 = 0$

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

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Step 6.4.1

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

$u - 2 = 0$

Step 6.4.2

Add $2$ to both sides of the equation.

$u = 2$

Step 6.5

The final solution is all the values that make $(u - 8) (u - 2) = 0$ true.

$u = 8, 2$

Step 7

Substitute $8$ for $u$ in $u = 4^{x}$ .

$8 = 4^{x}$

Step 8

Solve $8 = 4^{x}$ .

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Step 8.1

Rewrite the equation as $4^{x} = 8$ .

$4^{x} = 8$

Step 8.2

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

$2^{2 x} = 2^{3}$

Step 8.3

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

$2 x = 3$

Step 8.4

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 8.4.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $2$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 9

Substitute $2$ for $u$ in $u = 4^{x}$ .

$2 = 4^{x}$

Step 10

Solve $2 = 4^{x}$ .

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Step 10.1

Rewrite the equation as $4^{x} = 2$ .

$4^{x} = 2$

Step 10.2

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

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

Step 10.3

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

$2 x = 1$

Step 10.4

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 10.4.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

Cancel the common factor of $2$ .

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Step 10.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 10.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 11

List the solutions that makes the equation true.

$x = \frac{3}{2}, \frac{1}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}, \frac{1}{2}$

Decimal Form:

$x = 1.5, 0.5$

Mixed Number Form:

$x = 1 \frac{1}{2}, \frac{1}{2}$