Algebra Examples

$\frac{5 x + 7}{x - 2} - \frac{2 x + 21}{x + 2} = 8 \frac{2}{3}$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{5 x + 7}{x - 2} - \frac{2 x + 21}{x + 2}$ .

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Step 1.1.1

To write $\frac{5 x + 7}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{5 x + 7}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{2 x + 21}{x + 2} = 8 \frac{2}{3}$

Step 1.1.2

To write $- \frac{2 x + 21}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{5 x + 7}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{2 x + 21}{x + 2} \cdot \frac{x - 2}{x - 2} = 8 \frac{2}{3}$

Step 1.1.3

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

Multiply $\frac{5 x + 7}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{(5 x + 7) (x + 2)}{(x - 2) (x + 2)} - \frac{2 x + 21}{x + 2} \cdot \frac{x - 2}{x - 2} = 8 \frac{2}{3}$

Step 1.1.3.2

Multiply $\frac{2 x + 21}{x + 2}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(5 x + 7) (x + 2)}{(x - 2) (x + 2)} - \frac{(2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.3.3

Reorder the factors of $(x - 2) (x + 2)$ .

$\frac{(5 x + 7) (x + 2)}{(x + 2) (x - 2)} - \frac{(2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{(5 x + 7) (x + 2) - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Expand $(5 x + 7) (x + 2)$ using the FOIL Method.

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Step 1.1.5.1.1

Apply the distributive property.

$\frac{5 x (x + 2) + 7 (x + 2) - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.1.2

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot 2 + 7 (x + 2) - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.1.3

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot 2 + 7 x + 7 \cdot 2 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.2

Simplify and combine like terms.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

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

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Step 1.1.5.2.1.1.1

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot 2 + 7 x + 7 \cdot 2 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.2.1.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot 2 + 7 x + 7 \cdot 2 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.2.1.2

Multiply $2$ by $5$ .

$\frac{5 x^{2} + 10 x + 7 x + 7 \cdot 2 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.2.1.3

Multiply $7$ by $2$ .

$\frac{5 x^{2} + 10 x + 7 x + 14 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.2.2

Add $10 x$ and $7 x$ .

$\frac{5 x^{2} + 17 x + 14 - (2 x + 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.3

Apply the distributive property.

$\frac{5 x^{2} + 17 x + 14 + (- (2 x) - 1 \cdot 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.4

Multiply $2$ by $- 1$ .

$\frac{5 x^{2} + 17 x + 14 + (- 2 x - 1 \cdot 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.5

Multiply $- 1$ by $21$ .

$\frac{5 x^{2} + 17 x + 14 + (- 2 x - 21) (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.6

Expand $(- 2 x - 21) (x - 2)$ using the FOIL Method.

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Step 1.1.5.6.1

Apply the distributive property.

$\frac{5 x^{2} + 17 x + 14 - 2 x (x - 2) - 21 (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.6.2

Apply the distributive property.

$\frac{5 x^{2} + 17 x + 14 - 2 x \cdot x - 2 x \cdot - 2 - 21 (x - 2)}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.6.3

Apply the distributive property.

$\frac{5 x^{2} + 17 x + 14 - 2 x \cdot x - 2 x \cdot - 2 - 21 x - 21 \cdot - 2}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.7

Simplify and combine like terms.

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Step 1.1.5.7.1

Simplify each term.

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Step 1.1.5.7.1.1

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

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Step 1.1.5.7.1.1.1

Move $x$ .

$\frac{5 x^{2} + 17 x + 14 - 2 (x \cdot x) - 2 x \cdot - 2 - 21 x - 21 \cdot - 2}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.7.1.1.2

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 17 x + 14 - 2 x^{2} - 2 x \cdot - 2 - 21 x - 21 \cdot - 2}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.7.1.2

Multiply $- 2$ by $- 2$ .

$\frac{5 x^{2} + 17 x + 14 - 2 x^{2} + 4 x - 21 x - 21 \cdot - 2}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.7.1.3

Multiply $- 21$ by $- 2$ .

$\frac{5 x^{2} + 17 x + 14 - 2 x^{2} + 4 x - 21 x + 42}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.7.2

Subtract $21 x$ from $4 x$ .

$\frac{5 x^{2} + 17 x + 14 - 2 x^{2} - 17 x + 42}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.8

Subtract $2 x^{2}$ from $5 x^{2}$ .

$\frac{3 x^{2} + 17 x + 14 - 17 x + 42}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.9

Subtract $17 x$ from $17 x$ .

$\frac{3 x^{2} + 0 + 14 + 42}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.10

Add $3 x^{2}$ and $0$ .

$\frac{3 x^{2} + 14 + 42}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 1.1.5.11

Add $14$ and $42$ .

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = 8 \frac{2}{3}$

Step 2

Simplify the right side.

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Step 2.1

Convert $8 \frac{2}{3}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = 8 + \frac{2}{3}$

Step 2.1.2

Add $8$ and $\frac{2}{3}$ .

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Step 2.1.2.1

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = 8 \cdot \frac{3}{3} + \frac{2}{3}$

Step 2.1.2.2

Combine $8$ and $\frac{3}{3}$ .

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = \frac{8 \cdot 3}{3} + \frac{2}{3}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = \frac{8 \cdot 3 + 2}{3}$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $8$ by $3$ .

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = \frac{24 + 2}{3}$

Step 2.1.2.4.2

Add $24$ and $2$ .

$\frac{3 x^{2} + 56}{(x + 2) (x - 2)} = \frac{26}{3}$

Step 3

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.

$(3 x^{2} + 56) \cdot 3 = (x + 2) (x - 2) \cdot 26$

Step 4

Solve the equation for $x$ .

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Step 4.1

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

$(x + 2) (x - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2

Simplify $(x + 2) (x - 2) \cdot 26$ .

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Step 4.2.1

Rewrite.

$0 + 0 + (x + 2) (x - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.2

Simplify by adding zeros.

$(x + 2) (x - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.3

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

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Step 4.2.3.1

Apply the distributive property.

$(x (x - 2) + 2 (x - 2)) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.3.2

Apply the distributive property.

$(x \cdot x + x \cdot - 2 + 2 (x - 2)) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.3.3

Apply the distributive property.

$(x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4

Simplify terms.

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Step 4.2.4.1

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 4.2.4.1.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$(x \cdot x - 2 x + 2 x + 2 \cdot - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.1.2

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

$(x \cdot x + 0 + 2 \cdot - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.1.3

Add $x \cdot x$ and $0$ .

$(x \cdot x + 2 \cdot - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.2

Simplify each term.

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Step 4.2.4.2.1

Multiply $x$ by $x$ .

$(x^{2} + 2 \cdot - 2) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.2.2

Multiply $2$ by $- 2$ .

$(x^{2} - 4) \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.3

Simplify by multiplying through.

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Step 4.2.4.3.1

Apply the distributive property.

$x^{2} \cdot 26 - 4 \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.3.2

Simplify the expression.

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Step 4.2.4.3.2.1

Move $26$ to the left of $x^{2}$ .

$26 \cdot x^{2} - 4 \cdot 26 = (3 x^{2} + 56) \cdot 3$

Step 4.2.4.3.2.2

Multiply $- 4$ by $26$ .

$26 x^{2} - 104 = (3 x^{2} + 56) \cdot 3$

Step 4.3

Simplify $(3 x^{2} + 56) \cdot 3$ .

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Step 4.3.1

Apply the distributive property.

$26 x^{2} - 104 = 3 x^{2} \cdot 3 + 56 \cdot 3$

Step 4.3.2

Multiply.

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Step 4.3.2.1

Multiply $3$ by $3$ .

$26 x^{2} - 104 = 9 x^{2} + 56 \cdot 3$

Step 4.3.2.2

Multiply $56$ by $3$ .

$26 x^{2} - 104 = 9 x^{2} + 168$

Step 4.4

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

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Step 4.4.1

Subtract $9 x^{2}$ from both sides of the equation.

$26 x^{2} - 104 - 9 x^{2} = 168$

Step 4.4.2

Subtract $9 x^{2}$ from $26 x^{2}$ .

$17 x^{2} - 104 = 168$

Step 4.5

Move all terms not containing $x$ to the right side of the equation.

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Step 4.5.1

Add $104$ to both sides of the equation.

$17 x^{2} = 168 + 104$

Step 4.5.2

Add $168$ and $104$ .

$17 x^{2} = 272$

Step 4.6

Divide each term in $17 x^{2} = 272$ by $17$ and simplify.

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Step 4.6.1

Divide each term in $17 x^{2} = 272$ by $17$ .

$\frac{17 x^{2}}{17} = \frac{272}{17}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $17$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{17 x^{2}}{17} = \frac{272}{17}$

Step 4.6.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{272}{17}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Divide $272$ by $17$ .

$x^{2} = 16$

Step 4.7

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{16}$

Step 4.8

Simplify $\pm \sqrt{16}$ .

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Step 4.8.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 4.8.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 4$

Step 4.9

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.9.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4$

Step 4.9.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4$

Step 4.9.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4, - 4$