Precalculus Examples

$\frac{7}{2 - x} = \frac{10 - 4 x}{x^{2} + 3 x - 10}$

Step 1

Simplify both sides.

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Step 1.1

Factor $2$ out of $10 - 4 x$ .

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Step 1.1.1

Factor $2$ out of $10$ .

$\frac{7}{2 - x} = \frac{2 (5) - 4 x}{x^{2} + 3 x - 10}$

Step 1.1.2

Factor $2$ out of $- 4 x$ .

$\frac{7}{2 - x} = \frac{2 (5) + 2 (- 2 x)}{x^{2} + 3 x - 10}$

Step 1.1.3

Factor $2$ out of $2 (5) + 2 (- 2 x)$ .

$\frac{7}{2 - x} = \frac{2 (5 - 2 x)}{x^{2} + 3 x - 10}$

Step 1.2

Factor $x^{2} + 3 x - 10$ using the AC method.

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Step 1.2.1

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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{7}{2 - x} = \frac{2 (5 - 2 x)}{(x - 2) (x + 5)}$

Step 2

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.

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

Step 3

Solve the equation for $x$ .

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Step 3.1

Simplify $7 (x - 2) (x + 5)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 7 (x - 2) (x + 5) = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.2

Simplify by multiplying through.

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Step 3.1.2.1

Apply the distributive property.

$(7 x + 7 \cdot - 2) (x + 5) = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.2.2

Multiply $7$ by $- 2$ .

$(7 x - 14) (x + 5) = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.3

Expand $(7 x - 14) (x + 5)$ using the FOIL Method.

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Step 3.1.3.1

Apply the distributive property.

$7 x (x + 5) - 14 (x + 5) = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.3.2

Apply the distributive property.

$7 x \cdot x + 7 x \cdot 5 - 14 (x + 5) = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.3.3

Apply the distributive property.

$7 x \cdot x + 7 x \cdot 5 - 14 x - 14 \cdot 5 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.4

Simplify and combine like terms.

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Step 3.1.4.1

Simplify each term.

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Step 3.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.4.1.1.1

Move $x$ .

$7 (x \cdot x) + 7 x \cdot 5 - 14 x - 14 \cdot 5 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.4.1.1.2

Multiply $x$ by $x$ .

$7 x^{2} + 7 x \cdot 5 - 14 x - 14 \cdot 5 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.4.1.2

Multiply $5$ by $7$ .

$7 x^{2} + 35 x - 14 x - 14 \cdot 5 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.4.1.3

Multiply $- 14$ by $5$ .

$7 x^{2} + 35 x - 14 x - 70 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.1.4.2

Subtract $14 x$ from $35 x$ .

$7 x^{2} + 21 x - 70 = (2 - x) \cdot (2 (5 - 2 x))$

Step 3.2

Simplify $(2 - x) \cdot (2 (5 - 2 x))$ .

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Step 3.2.1

Simplify by multiplying through.

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Step 3.2.1.1

Apply the distributive property.

$7 x^{2} + 21 x - 70 = (2 - x) \cdot (2 \cdot 5 + 2 (- 2 x))$

Step 3.2.1.2

Multiply.

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Step 3.2.1.2.1

Multiply $2$ by $5$ .

$7 x^{2} + 21 x - 70 = (2 - x) \cdot (10 + 2 (- 2 x))$

Step 3.2.1.2.2

Multiply $- 2$ by $2$ .

$7 x^{2} + 21 x - 70 = (2 - x) \cdot (10 - 4 x)$

Step 3.2.2

Expand $(2 - x) (10 - 4 x)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$7 x^{2} + 21 x - 70 = 2 (10 - 4 x) - x (10 - 4 x)$

Step 3.2.2.2

Apply the distributive property.

$7 x^{2} + 21 x - 70 = 2 \cdot 10 + 2 (- 4 x) - x (10 - 4 x)$

Step 3.2.2.3

Apply the distributive property.

$7 x^{2} + 21 x - 70 = 2 \cdot 10 + 2 (- 4 x) - x \cdot 10 - x (- 4 x)$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $2$ by $10$ .

$7 x^{2} + 21 x - 70 = 20 + 2 (- 4 x) - x \cdot 10 - x (- 4 x)$

Step 3.2.3.1.2

Multiply $- 4$ by $2$ .

$7 x^{2} + 21 x - 70 = 20 - 8 x - x \cdot 10 - x (- 4 x)$

Step 3.2.3.1.3

Multiply $10$ by $- 1$ .

$7 x^{2} + 21 x - 70 = 20 - 8 x - 10 x - x (- 4 x)$

Step 3.2.3.1.4

Rewrite using the commutative property of multiplication.

$7 x^{2} + 21 x - 70 = 20 - 8 x - 10 x - 1 \cdot - 4 x \cdot x$

Step 3.2.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.3.1.5.1

Move $x$ .

$7 x^{2} + 21 x - 70 = 20 - 8 x - 10 x - 1 \cdot - 4 (x \cdot x)$

Step 3.2.3.1.5.2

Multiply $x$ by $x$ .

$7 x^{2} + 21 x - 70 = 20 - 8 x - 10 x - 1 \cdot - 4 x^{2}$

Step 3.2.3.1.6

Multiply $- 1$ by $- 4$ .

$7 x^{2} + 21 x - 70 = 20 - 8 x - 10 x + 4 x^{2}$

Step 3.2.3.2

Subtract $10 x$ from $- 8 x$ .

$7 x^{2} + 21 x - 70 = 20 - 18 x + 4 x^{2}$

Step 3.3

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

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Step 3.3.1

Add $18 x$ to both sides of the equation.

$7 x^{2} + 21 x - 70 + 18 x = 20 + 4 x^{2}$

Step 3.3.2

Subtract $4 x^{2}$ from both sides of the equation.

$7 x^{2} + 21 x - 70 + 18 x - 4 x^{2} = 20$

Step 3.3.3

Subtract $4 x^{2}$ from $7 x^{2}$ .

$3 x^{2} + 21 x - 70 + 18 x = 20$

Step 3.3.4

Add $21 x$ and $18 x$ .

$3 x^{2} + 39 x - 70 = 20$

Step 3.4

Subtract $20$ from both sides of the equation.

$3 x^{2} + 39 x - 70 - 20 = 0$

Step 3.5

Subtract $20$ from $- 70$ .

$3 x^{2} + 39 x - 90 = 0$

Step 3.6

Factor the left side of the equation.

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Step 3.6.1

Factor $3$ out of $3 x^{2} + 39 x - 90$ .

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Step 3.6.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 39 x - 90 = 0$

Step 3.6.1.2

Factor $3$ out of $39 x$ .

$3 (x^{2}) + 3 (13 x) - 90 = 0$

Step 3.6.1.3

Factor $3$ out of $- 90$ .

$3 x^{2} + 3 (13 x) + 3 \cdot - 30 = 0$

Step 3.6.1.4

Factor $3$ out of $3 x^{2} + 3 (13 x)$ .

$3 (x^{2} + 13 x) + 3 \cdot - 30 = 0$

Step 3.6.1.5

Factor $3$ out of $3 (x^{2} + 13 x) + 3 \cdot - 30$ .

$3 (x^{2} + 13 x - 30) = 0$

Step 3.6.2

Factor.

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Step 3.6.2.1

Factor $x^{2} + 13 x - 30$ using the AC method.

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Step 3.6.2.1.1

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 $- 30$ and whose sum is $13$ .

$- 2, 15$

Step 3.6.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 15)) = 0$

Step 3.6.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 15) = 0$

Step 3.7

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

$x - 2 = 0$

$x + 15 = 0$

Step 3.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.9

Set $x + 15$ equal to $0$ and solve for $x$ .

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Step 3.9.1

Set $x + 15$ equal to $0$ .

$x + 15 = 0$

Step 3.9.2

Subtract $15$ from both sides of the equation.

$x = - 15$

Step 3.10

The final solution is all the values that make $3 (x - 2) (x + 15) = 0$ true.

$x = 2, - 15$

Step 4

Exclude the solutions that do not make $\frac{7}{2 - x} = \frac{10 - 4 x}{x^{2} + 3 x - 10}$ true.

$x = - 15$