Trigonometry Examples

$\frac{1}{x + 3} = \frac{x + 10}{x - 2}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$1 (x - 2) = (x + 3) (x + 10)$

Step 2

Solve the equation for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(x + 3) (x + 10) = 1 (x - 2)$

Step 2.2

Simplify $(x + 3) (x + 10)$ .

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Step 2.2.1

Rewrite.

$0 + 0 + (x + 3) (x + 10) = 1 (x - 2)$

Step 2.2.2

Simplify by adding zeros.

$(x + 3) (x + 10) = 1 (x - 2)$

Step 2.2.3

Expand $(x + 3) (x + 10)$ using the FOIL Method.

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Step 2.2.3.1

Apply the distributive property.

$x (x + 10) + 3 (x + 10) = 1 (x - 2)$

Step 2.2.3.2

Apply the distributive property.

$x \cdot x + x \cdot 10 + 3 (x + 10) = 1 (x - 2)$

Step 2.2.3.3

Apply the distributive property.

$x \cdot x + x \cdot 10 + 3 x + 3 \cdot 10 = 1 (x - 2)$

Step 2.2.4

Simplify and combine like terms.

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 10 + 3 x + 3 \cdot 10 = 1 (x - 2)$

Step 2.2.4.1.2

Move $10$ to the left of $x$ .

$x^{2} + 10 \cdot x + 3 x + 3 \cdot 10 = 1 (x - 2)$

Step 2.2.4.1.3

Multiply $3$ by $10$ .

$x^{2} + 10 x + 3 x + 30 = 1 (x - 2)$

Step 2.2.4.2

Add $10 x$ and $3 x$ .

$x^{2} + 13 x + 30 = 1 (x - 2)$

Step 2.3

Multiply $x - 2$ by $1$ .

$x^{2} + 13 x + 30 = x - 2$

Step 2.4

Move all terms containing $x$ to the left side of the equation.

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Step 2.4.1

Subtract $x$ from both sides of the equation.

$x^{2} + 13 x + 30 - x = - 2$

Step 2.4.2

Subtract $x$ from $13 x$ .

$x^{2} + 12 x + 30 = - 2$

Step 2.5

Add $2$ to both sides of the equation.

$x^{2} + 12 x + 30 + 2 = 0$

Step 2.6

Add $30$ and $2$ .

$x^{2} + 12 x + 32 = 0$

Step 2.7

Factor $x^{2} + 12 x + 32$ using the AC method.

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Step 2.7.1

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 $32$ and whose sum is $12$ .

$4, 8$

Step 2.7.2

Write the factored form using these integers.

$(x + 4) (x + 8) = 0$

Step 2.8

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

$x + 4 = 0$

$x + 8 = 0$

Step 2.9

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.9.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.9.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.10

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 2.10.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 2.10.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.11

The final solution is all the values that make $(x + 4) (x + 8) = 0$ true.

$x = - 4, - 8$