Trigonometry Examples

$9^{8 x - 32} = 3^{x^{2}}$

Step 1

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

$3^{2 (8 x - 32)} = 3^{x^{2}}$

Step 2

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

$2 (8 x - 32) = x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (8 x - 32)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (8 x - 32) = x^{2}$

Step 3.1.2

Simplify by adding zeros.

$2 (8 x - 32) = x^{2}$

Step 3.1.3

Apply the distributive property.

$2 (8 x) + 2 \cdot - 32 = x^{2}$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $8$ by $2$ .

$16 x + 2 \cdot - 32 = x^{2}$

Step 3.1.4.2

Multiply $2$ by $- 32$ .

$16 x - 64 = x^{2}$

Step 3.2

Subtract $x^{2}$ from both sides of the equation.

$16 x - 64 - x^{2} = 0$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $- 1$ out of $16 x - 64 - x^{2}$ .

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Step 3.3.1.1

Reorder the expression.

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Step 3.3.1.1.1

Move $- 64$ .

$16 x - x^{2} - 64 = 0$

Step 3.3.1.1.2

Reorder $16 x$ and $- x^{2}$ .

$- x^{2} + 16 x - 64 = 0$

Step 3.3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 16 x - 64 = 0$

Step 3.3.1.3

Factor $- 1$ out of $16 x$ .

$- (x^{2}) - (- 16 x) - 64 = 0$

Step 3.3.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$- (x^{2}) - (- 16 x) - 1 \cdot 64 = 0$

Step 3.3.1.5

Factor $- 1$ out of $- (x^{2}) - (- 16 x)$ .

$- (x^{2} - 16 x) - 1 \cdot 64 = 0$

Step 3.3.1.6

Factor $- 1$ out of $- (x^{2} - 16 x) - 1 (64)$ .

$- (x^{2} - 16 x + 64) = 0$

Step 3.3.2

Factor using the perfect square rule.

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Step 3.3.2.1

Rewrite $64$ as $8^{2}$ .

$- (x^{2} - 16 x + 8^{2}) = 0$

Step 3.3.2.2

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

$16 x = 2 \cdot x \cdot 8$

Step 3.3.2.3

Rewrite the polynomial.

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

Step 3.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 8$ .

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

Step 3.4

Divide each term in $- {(x - 8)}^{2} = 0$ by $- 1$ and simplify.

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Step 3.4.1

Divide each term in $- {(x - 8)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 8)}^{2}}{- 1} = \frac{0}{- 1}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 8)}^{2}}{1} = \frac{0}{- 1}$

Step 3.4.2.2

Divide ${(x - 8)}^{2}$ by $1$ .

${(x - 8)}^{2} = \frac{0}{- 1}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $0$ by $- 1$ .

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

Step 3.5

Set the $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 3.6

Add $8$ to both sides of the equation.

$x = 8$