Trigonometry Examples

${(\frac{1}{5})}^{- 2 x^{2} - 4} = {(\frac{1}{125})}^{- 8 x + 22}$

Step 1

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

${(\frac{1}{5})}^{- 2 x^{2} - 4} = {(\frac{1}{5})}^{3 (- 8 x + 22)}$

Step 2

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

$- 2 x^{2} - 4 = 3 (- 8 x + 22)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (- 8 x + 22)$ .

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Step 3.1.1

Rewrite.

$- 2 x^{2} - 4 = 0 + 0 + 3 (- 8 x + 22)$

Step 3.1.2

Simplify by adding zeros.

$- 2 x^{2} - 4 = 3 (- 8 x + 22)$

Step 3.1.3

Apply the distributive property.

$- 2 x^{2} - 4 = 3 (- 8 x) + 3 \cdot 22$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $- 8$ by $3$ .

$- 2 x^{2} - 4 = - 24 x + 3 \cdot 22$

Step 3.1.4.2

Multiply $3$ by $22$ .

$- 2 x^{2} - 4 = - 24 x + 66$

Step 3.2

Add $24 x$ to both sides of the equation.

$- 2 x^{2} - 4 + 24 x = 66$

Step 3.3

Subtract $66$ from both sides of the equation.

$- 2 x^{2} - 4 + 24 x - 66 = 0$

Step 3.4

Subtract $66$ from $- 4$ .

$- 2 x^{2} + 24 x - 70 = 0$

Step 3.5

Factor the left side of the equation.

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Step 3.5.1

Factor $- 2$ out of $- 2 x^{2} + 24 x - 70$ .

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Step 3.5.1.1

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

$- 2 x^{2} + 24 x - 70 = 0$

Step 3.5.1.2

Factor $- 2$ out of $24 x$ .

$- 2 x^{2} - 2 (- 12 x) - 70 = 0$

Step 3.5.1.3

Factor $- 2$ out of $- 70$ .

$- 2 x^{2} - 2 (- 12 x) - 2 \cdot 35 = 0$

Step 3.5.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 12 x)$ .

$- 2 (x^{2} - 12 x) - 2 \cdot 35 = 0$

Step 3.5.1.5

Factor $- 2$ out of $- 2 (x^{2} - 12 x) - 2 (35)$ .

$- 2 (x^{2} - 12 x + 35) = 0$

Step 3.5.2

Factor.

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Step 3.5.2.1

Factor $x^{2} - 12 x + 35$ using the AC method.

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Step 3.5.2.1.1

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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 3.5.2.1.2

Write the factored form using these integers.

$- 2 ((x - 7) (x - 5)) = 0$

Step 3.5.2.2

Remove unnecessary parentheses.

$- 2 (x - 7) (x - 5) = 0$

Step 3.6

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

$x - 7 = 0$

$x - 5 = 0$

Step 3.7

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.7.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.8

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.8.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3.9

The final solution is all the values that make $- 2 (x - 7) (x - 5) = 0$ true.

$x = 7, 5$