Algebra Examples

$4^{x + 1} + 2^{x + 3} = 320$

Step 1

Subtract $320$ from both sides of the equation.

$4^{x + 1} + 2^{x + 3} - 320 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

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

$4^{x} \cdot 4 + 2^{x + 3} - 320 = 0$

Step 2.2

Rewrite $2^{x + 3}$ as $2^{x} \cdot 2^{3}$ .

$4^{x} \cdot 4 + 2^{x} \cdot 2^{3} - 320 = 0$

Step 2.3

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

${(2^{2})}^{x} \cdot 4 + 2^{x} \cdot 2^{3} - 320 = 0$

Step 2.4

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

${(2^{x})}^{2} \cdot 4 + 2^{x} \cdot 2^{3} - 320 = 0$

Step 2.5

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

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Step 2.5.1

Evaluate the exponent.

$u^{2} \cdot 4 + u \cdot 2^{3} - 320 = 0$

Step 2.5.2

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

$4 \cdot u^{2} + u \cdot 2^{3} - 320 = 0$

Step 2.5.3

Raise $2$ to the power of $3$ .

$4 u^{2} + u \cdot 8 - 320 = 0$

Step 2.5.4

Move $8$ to the left of $u$ .

$4 u^{2} + 8 u - 320 = 0$

Step 2.6

Factor $4$ out of $4 u^{2} + 8 u - 320$ .

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Step 2.6.1

Factor $4$ out of $4 u^{2}$ .

$4 (u^{2}) + 8 u - 320 = 0$

Step 2.6.2

Factor $4$ out of $8 u$ .

$4 (u^{2}) + 4 (2 u) - 320 = 0$

Step 2.6.3

Factor $4$ out of $- 320$ .

$4 u^{2} + 4 (2 u) + 4 \cdot - 80 = 0$

Step 2.6.4

Factor $4$ out of $4 u^{2} + 4 (2 u)$ .

$4 (u^{2} + 2 u) + 4 \cdot - 80 = 0$

Step 2.6.5

Factor $4$ out of $4 (u^{2} + 2 u) + 4 \cdot - 80$ .

$4 (u^{2} + 2 u - 80) = 0$

Step 2.7

Factor.

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Step 2.7.1

Factor $u^{2} + 2 u - 80$ using the AC method.

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Step 2.7.1.1

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 $- 80$ and whose sum is $2$ .

$- 8, 10$

Step 2.7.1.2

Write the factored form using these integers.

$4 ((u - 8) (u + 10)) = 0$

Step 2.7.2

Remove unnecessary parentheses.

$4 (u - 8) (u + 10) = 0$

Step 2.8

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

$4 (2^{x} - 8) (2^{x} + 10) = 0$

Step 2.9

Simplify.

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Step 2.9.1

Apply the distributive property.

$(4 \cdot 2^{x} + 4 \cdot - 8) (2^{x} + 10) = 0$

Step 2.9.2

Multiply $4 \cdot 2^{x}$ .

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Step 2.9.2.1

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

$(2^{2} \cdot 2^{x} + 4 \cdot - 8) (2^{x} + 10) = 0$

Step 2.9.2.2

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

$(2^{2 + x} + 4 \cdot - 8) (2^{x} + 10) = 0$

Step 2.9.3

Multiply $4$ by $- 8$ .

$(2^{2 + x} - 32) (2^{x} + 10) = 0$

Step 3

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

$2^{2 + x} - 32 = 0$

$2^{x} + 10 = 0$

Step 4

Set $2^{2 + x} - 32$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2^{2 + x} - 32$ equal to $0$ .

$2^{2 + x} - 32 = 0$

Step 4.2

Solve $2^{2 + x} - 32 = 0$ for $x$ .

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Step 4.2.1

Add $32$ to both sides of the equation.

$2^{2 + x} = 32$

Step 4.2.2

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

$2^{2 + x} = 2^{5}$

Step 4.2.3

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

$2 + x = 5$

Step 4.2.4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.4.1

Subtract $2$ from both sides of the equation.

$x = 5 - 2$

Step 4.2.4.2

Subtract $2$ from $5$ .

$x = 3$

Step 5

Set $2^{x} + 10$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2^{x} + 10$ equal to $0$ .

$2^{x} + 10 = 0$

Step 5.2

Solve $2^{x} + 10 = 0$ for $x$ .

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Step 5.2.1

Subtract $10$ from both sides of the equation.

$2^{x} = - 10$

Step 5.2.2

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

$\ln (2^{x}) = \ln (- 10)$

Step 5.2.3

The equation cannot be solved because $\ln (- 10)$ is undefined.

Undefined

Step 5.2.4

There is no solution for $2^{x} = - 10$

No solution

Step 6

The final solution is all the values that make $(2^{2 + x} - 32) (2^{x} + 10) = 0$ true.

$x = 3$