Find All Complex Solutions 3^(x+1)+3^(3-x)=30

$3^{x + 1} + 3^{3 - x} = 30$

Step 1

Rewrite $3^{x + 1}$ as $3^{x} \cdot 3^{1}$ .

$3^{x} \cdot 3 + 3^{3 - x} = 30$

Step 2

Rewrite $3^{3 - x}$ as $3^{3} \cdot 3^{- x}$ .

$3^{x} \cdot 3 + 3^{3} \cdot 3^{- x} = 30$

Step 3

Rewrite $3^{- x}$ as exponentiation.

$3^{x} \cdot 3 + 3^{3} \cdot {(3^{x})}^{- 1} = 30$

Step 4

Substitute $u$ for $3^{x}$ .

$u \cdot 3 + 3^{3} \cdot u^{- 1} = 30$

Step 5

Simplify each term.

Move $3$ to the left of $u$ .

$3 \cdot u + 3^{3} \cdot u^{- 1} = 30$

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

$3 u + 27 \cdot u^{- 1} = 30$

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

$3 u + 27 \cdot \frac{1}{u} = 30$

Combine $27$ and $\frac{1}{u}$ .

$3 u + \frac{27}{u} = 30$

Step 6

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 $3 u + \frac{27}{u} = 30$ by $u$ to eliminate the fractions.

Multiply each term in $3 u + \frac{27}{u} = 30$ by $u$ .

$3 u \cdot u + \frac{27}{u} u = 30 u$

Simplify the left side.

Simplify each term.

Multiply $u$ by $u$ by adding the exponents.

Move $u$ .

$3 (u \cdot u) + \frac{27}{u} u = 30 u$

Multiply $u$ by $u$ .

$3 u^{2} + \frac{27}{u} u = 30 u$

Cancel the common factor of $u$ .

Cancel the common factor.

$3 u^{2} + \frac{27}{u} u = 30 u$

Rewrite the expression.

$3 u^{2} + 27 = 30 u$

Solve the equation.

Subtract $30 u$ from both sides of the equation.

$3 u^{2} + 27 - 30 u = 0$

Factor the left side of the equation.

Factor $3$ out of $3 u^{2} + 27 - 30 u$ .

Factor $3$ out of $3 u^{2}$ .

$3 (u^{2}) + 27 - 30 u = 0$

Factor $3$ out of $27$ .

$3 u^{2} + 3 \cdot 9 - 30 u = 0$

Factor $3$ out of $- 30 u$ .

$3 u^{2} + 3 \cdot 9 + 3 (- 10 u) = 0$

Factor $3$ out of $3 u^{2} + 3 \cdot 9$ .

$3 (u^{2} + 9) + 3 (- 10 u) = 0$

Factor $3$ out of $3 (u^{2} + 9) + 3 (- 10 u)$ .

$3 (u^{2} + 9 - 10 u) = 0$

Factor.

Factor $u^{2} + 9 - 10 u$ 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Write the factored form using these integers.

$3 ((u - 9) (u - 1)) = 0$

Remove unnecessary parentheses.

$3 (u - 9) (u - 1) = 0$

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

$u - 9 = 0$

$u - 1 = 0$

Set $u - 9$ equal to $0$ and solve for $u$ .

Set $u - 9$ equal to $0$ .

$u - 9 = 0$

Add $9$ to both sides of the equation.

$u = 9$

Set $u - 1$ equal to $0$ and solve for $u$ .

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Add $1$ to both sides of the equation.

$u = 1$

The final solution is all the values that make $3 (u - 9) (u - 1) = 0$ true.

$u = 9, 1$

Step 7

Substitute $9$ for $u$ in $u = 3^{x}$ .

$9 = 3^{x}$

Step 8

Solve $9 = 3^{x}$ .

Rewrite the equation as $3^{x} = 9$ .

$3^{x} = 9$

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

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

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

$x = 2$

Step 9

Substitute $1$ for $u$ in $u = 3^{x}$ .

$1 = 3^{x}$

Step 10

Solve $1 = 3^{x}$ .

Rewrite the equation as $3^{x} = 1$ .

$3^{x} = 1$

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

$\ln (3^{x}) = \ln (1)$

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (1)$

Simplify the right side.

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

$x \ln (3) = 0$

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ and simplify.

Divide each term in $x \ln (3) = 0$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Simplify the left side.

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{0}{\ln (3)}$

Divide $x$ by $1$ .

$x = \frac{0}{\ln (3)}$

Simplify the right side.

Divide $0$ by $\ln (3)$ .

$x = 0$

Step 11

List the solutions that makes the equation true.

$x = 2, 0$

Find All Complex Solutions 3^(x+1)+3^(3-x)=30

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