Algebra Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{- 2}{x - 7}$ from both sides of the equation.

$2 - \frac{- 2}{x - 7} = \frac{4}{{(x - 7)}^{2}}$

Step 1.2

Subtract $\frac{4}{{(x - 7)}^{2}}$ from both sides of the equation.

$2 - \frac{- 2}{x - 7} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2

Simplify $2 - \frac{- 2}{x - 7} - \frac{4}{{(x - 7)}^{2}}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Write $2$ as a fraction with denominator $1$ .

$\frac{2}{1} - \frac{- 2}{x - 7} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.2

Multiply $\frac{2}{1}$ by $\frac{{(x - 7)}^{2}}{{(x - 7)}^{2}}$ .

$\frac{2}{1} \cdot \frac{{(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2}{x - 7} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.3

Multiply $\frac{2}{1}$ by $\frac{{(x - 7)}^{2}}{{(x - 7)}^{2}}$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2}{x - 7} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.4

Multiply $\frac{- 2}{x - 7}$ by $\frac{x - 7}{x - 7}$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - (\frac{- 2}{x - 7} \cdot \frac{x - 7}{x - 7}) - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.5

Multiply $\frac{- 2}{x - 7}$ by $\frac{x - 7}{x - 7}$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2 (x - 7)}{(x - 7) (x - 7)} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.6

Raise $x - 7$ to the power of $1$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2 (x - 7)}{{(x - 7)}^{1} (x - 7)} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.7

Raise $x - 7$ to the power of $1$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2 (x - 7)}{{(x - 7)}^{1} {(x - 7)}^{1}} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2 (x - 7)}{{(x - 7)}^{1 + 1}} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.1.9

Add $1$ and $1$ .

$\frac{2 {(x - 7)}^{2}}{{(x - 7)}^{2}} - \frac{- 2 (x - 7)}{{(x - 7)}^{2}} - \frac{4}{{(x - 7)}^{2}} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 {(x - 7)}^{2} + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$\frac{2 ((x - 7) (x - 7)) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

$\frac{2 (x (x - 7) - 7 (x - 7)) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.2.2

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot - 7 - 7 (x - 7)) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.2.3

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.3.1.2

Move $- 7$ to the left of $x$ .

$\frac{2 (x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.3.1.3

Multiply $- 7$ by $- 7$ .

$\frac{2 (x^{2} - 7 x - 7 x + 49) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.3.2

Subtract $7 x$ from $- 7 x$ .

$\frac{2 (x^{2} - 14 x + 49) + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{2 x^{2} + 2 (- 14 x) + 2 \cdot 49 + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply $- 14$ by $2$ .

$\frac{2 x^{2} - 28 x + 2 \cdot 49 + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.5.2

Multiply $2$ by $49$ .

$\frac{2 x^{2} - 28 x + 98 + 2 (x - 7) - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.6

Apply the distributive property.

$\frac{2 x^{2} - 28 x + 98 + 2 x + 2 \cdot - 7 - 4}{{(x - 7)}^{2}} = 0$

Step 2.3.7

Multiply $2$ by $- 7$ .

$\frac{2 x^{2} - 28 x + 98 + 2 x - 14 - 4}{{(x - 7)}^{2}} = 0$

Step 2.4

Add $- 28 x$ and $2 x$ .

$\frac{2 x^{2} - 26 x + 98 - 14 - 4}{{(x - 7)}^{2}} = 0$

Step 2.5

Subtract $14$ from $98$ .

$\frac{2 x^{2} - 26 x + 84 - 4}{{(x - 7)}^{2}} = 0$

Step 2.6

Subtract $4$ from $84$ .

$\frac{2 x^{2} - 26 x + 80}{{(x - 7)}^{2}} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Factor $2$ out of $2 x^{2} - 26 x + 80$ .

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Step 2.7.1.1

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

$\frac{2 (x^{2}) - 26 x + 80}{{(x - 7)}^{2}} = 0$

Step 2.7.1.2

Factor $2$ out of $- 26 x$ .

$\frac{2 (x^{2}) + 2 (- 13 x) + 80}{{(x - 7)}^{2}} = 0$

Step 2.7.1.3

Factor $2$ out of $80$ .

$\frac{2 x^{2} + 2 (- 13 x) + 2 \cdot 40}{{(x - 7)}^{2}} = 0$

Step 2.7.1.4

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

$\frac{2 (x^{2} - 13 x) + 2 \cdot 40}{{(x - 7)}^{2}} = 0$

Step 2.7.1.5

Factor $2$ out of $2 (x^{2} - 13 x) + 2 \cdot 40$ .

$\frac{2 (x^{2} - 13 x + 40)}{{(x - 7)}^{2}} = 0$

Step 2.7.2

Factor $x^{2} - 13 x + 40$ using the AC method.

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Step 2.7.2.1

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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Step 2.7.2.2

Write the factored form using these integers.

$\frac{2 ((x - 8) (x - 5))}{{(x - 7)}^{2}} = 0$

$\frac{2 (x - 8) (x - 5)}{{(x - 7)}^{2}} = 0$

Step 3

Set the numerator equal to zero.

$2 (x - 8) (x - 5) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

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

$x - 8 = 0$

$x - 5 = 0$

Step 4.2

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

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Step 4.2.1

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

$x - 8 = 0$

Step 4.2.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4.3

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

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Step 4.3.1

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

$x - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.4

The final solution is all the values that make $2 (x - 8) (x - 5) = 0$ true.

$x = 8, 5$