Solve for x 3^x+3^(3-x)=12

$3^{x} + 3^{3 - x} = 12$

Step 1

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

$3^{x} + 3^{3} \cdot 3^{- x} = 12$

Step 2

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

$3^{x} + 3^{3} \cdot {(3^{x})}^{- 1} = 12$

Step 3

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

$u + 3^{3} \cdot u^{- 1} = 12$

Step 4

Simplify each term.

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

$u + 27 \cdot u^{- 1} = 12$

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

$u + 27 \cdot \frac{1}{u} = 12$

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

$u + \frac{27}{u} = 12$

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 $u + \frac{27}{u} = 12$ by $u$ to eliminate the fractions.

Multiply each term in $u + \frac{27}{u} = 12$ by $u$ .

$u \cdot u + \frac{27}{u} u = 12 u$

Simplify the left side.

Simplify each term.

Multiply $u$ by $u$ .

$u^{2} + \frac{27}{u} u = 12 u$

Cancel the common factor of $u$ .

Cancel the common factor.

$u^{2} + \frac{27}{u} u = 12 u$

Rewrite the expression.

$u^{2} + 27 = 12 u$

Solve the equation.

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

$u^{2} + 27 - 12 u = 0$

Factor $u^{2} + 27 - 12 u$ 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 $27$ and whose sum is $- 12$ .

$- 9, - 3$

Write the factored form using these integers.

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

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

$u - 9 = 0$

$u - 3 = 0$

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 $u - 3$ equal to $0$ and solve for $u$ .

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Add $3$ to both sides of the equation.

$u = 3$

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

$u = 9, 3$

Step 6

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

$9 = 3^{x}$

Step 7

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 8

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

$3 = 3^{x}$

Step 9

Solve $3 = 3^{x}$ .

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

$3^{x} = 3$

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

$3^{x} = 3^{1}$

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

$x = 1$

Step 10

List the solutions that makes the equation true.

$x = 2, 1$

Solve for x 3^x+3^(3-x)=12

Related Questions

Support LCMGCF.com by disabling your adblocker.