Find All Complex Solutions 2^x+2^(5-x)=12

$2^{x} + 2^{5 - x} = 12$

Step 1

Rewrite $2^{5 - x}$ as $2^{5} \cdot 2^{- x}$ .

$2^{x} + 2^{5} \cdot 2^{- x} = 12$

Step 2

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

$2^{x} + 2^{5} \cdot {(2^{x})}^{- 1} = 12$

Step 3

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

$u + 2^{5} \cdot u^{- 1} = 12$

Step 4

Simplify each term.

Raise $2$ to the power of $5$ .

$u + 32 \cdot u^{- 1} = 12$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u + 32 \cdot \frac{1}{u} = 12$

Combine $32$ and $\frac{1}{u}$ .

$u + \frac{32}{u} = 12$

Step 5

Solve for $u$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1$

The LCM of one and any expression is the expression.

$u$

Multiply each term in $u + \frac{32}{u} = 12$ by $u$ to eliminate the fractions.

Multiply each term in $u + \frac{32}{u} = 12$ by $u$ .

$u \cdot u + \frac{32}{u} u = 12 u$

Simplify the left side.

Simplify each term.

Multiply $u$ by $u$ .

$u^{2} + \frac{32}{u} u = 12 u$

Cancel the common factor of $u$ .

Cancel the common factor.

$u^{2} + \frac{32}{u} u = 12 u$

Rewrite the expression.

$u^{2} + 32 = 12 u$

Solve the equation.

Subtract $12 u$ from both sides of the equation.

$u^{2} + 32 - 12 u = 0$

Factor $u^{2} + 32 - 12 u$ using the AC method.

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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Write the factored form using these integers.

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

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

$u - 8 = 0$

$u - 4 = 0$

Set $u - 8$ equal to $0$ and solve for $u$ .

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

Add $8$ to both sides of the equation.

$u = 8$

Set $u - 4$ equal to $0$ and solve for $u$ .

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Add $4$ to both sides of the equation.

$u = 4$

The final solution is all the values that make $(u - 8) (u - 4) = 0$ true.

$u = 8, 4$

Step 6

Substitute $8$ for $u$ in $u = 2^{x}$ .

$8 = 2^{x}$

Step 7

Solve $8 = 2^{x}$ .

Rewrite the equation as $2^{x} = 8$ .

$2^{x} = 8$

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

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

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

$x = 3$

Step 8

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

$4 = 2^{x}$

Step 9

Solve $4 = 2^{x}$ .

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

$2^{x} = 4$

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

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

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

$x = 2$

Step 10

List the solutions that makes the equation true.

$x = 3, 2$

Find All Complex Solutions 2^x+2^(5-x)=12

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