Calculus Examples

$3^{2 x} - 18 \cdot 3^{x} + 81 = 0$

Step 1

Rewrite $3^{2 x}$ as exponentiation.

${(3^{x})}^{2} - 18 \cdot 3^{x} + 81 = 0$

Step 2

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

$u^{2} - 18 u + 81 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Factor using the perfect square rule.

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Step 3.1.1

Rewrite $81$ as $9^{2}$ .

$u^{2} - 18 u + 9^{2} = 0$

Step 3.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 u = 2 \cdot u \cdot 9$

Step 3.1.3

Rewrite the polynomial.

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

Step 3.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 9$ .

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

Step 3.2

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

$u - 9 = 0$

Step 3.3

Add $9$ to both sides of the equation.

$u = 9$

Step 4

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

$9 = 3^{x}$

Step 5

Solve $9 = 3^{x}$ .

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Step 5.1

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

$3^{x} = 9$

Step 5.2

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

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

Step 5.3

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

$x = 2$