Algebra Examples

$2 (\frac{2 x + 3}{x - 3}) - 25 \frac{x - 3}{2 x + 3} = 5$

Step 1

Simplify $2 (\frac{2 x + 3}{x - 3}) - 25 \frac{x - 3}{2 x + 3}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $2$ and $\frac{2 x + 3}{x - 3}$ .

$\frac{2 (2 x + 3)}{x - 3} - 25 \frac{x - 3}{2 x + 3} = 5$

Step 1.1.2

Combine $- 25$ and $\frac{x - 3}{2 x + 3}$ .

$\frac{2 (2 x + 3)}{x - 3} + \frac{- 25 (x - 3)}{2 x + 3} = 5$

Step 1.1.3

Move the negative in front of the fraction.

$\frac{2 (2 x + 3)}{x - 3} - \frac{25 (x - 3)}{2 x + 3} = 5$

Step 1.2

To write $\frac{2 (2 x + 3)}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2 x + 3}{2 x + 3}$ .

$\frac{2 (2 x + 3)}{x - 3} \cdot \frac{2 x + 3}{2 x + 3} - \frac{25 (x - 3)}{2 x + 3} = 5$

Step 1.3

To write $- \frac{25 (x - 3)}{2 x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{2 (2 x + 3)}{x - 3} \cdot \frac{2 x + 3}{2 x + 3} - \frac{25 (x - 3)}{2 x + 3} \cdot \frac{x - 3}{x - 3} = 5$

Step 1.4

Write each expression with a common denominator of $(x - 3) (2 x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 (2 x + 3)}{x - 3}$ by $\frac{2 x + 3}{2 x + 3}$ .

$\frac{2 (2 x + 3) (2 x + 3)}{(x - 3) (2 x + 3)} - \frac{25 (x - 3)}{2 x + 3} \cdot \frac{x - 3}{x - 3} = 5$

Step 1.4.2

Multiply $\frac{25 (x - 3)}{2 x + 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{2 (2 x + 3) (2 x + 3)}{(x - 3) (2 x + 3)} - \frac{25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.4.3

Reorder the factors of $(x - 3) (2 x + 3)$ .

$\frac{2 (2 x + 3) (2 x + 3)}{(2 x + 3) (x - 3)} - \frac{25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 (2 x + 3) (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{(2 (2 x) + 2 \cdot 3) (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{(4 x + 2 \cdot 3) (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.3

Multiply $2$ by $3$ .

$\frac{(4 x + 6) (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.4

Expand $(4 x + 6) (2 x + 3)$ using the FOIL Method.

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Step 1.6.4.1

Apply the distributive property.

$\frac{4 x (2 x + 3) + 6 (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.4.2

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot 3 + 6 (2 x + 3) - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.4.3

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot 3 + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5

Simplify and combine like terms.

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Step 1.6.5.1

Simplify each term.

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Step 1.6.5.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot 2 x \cdot x + 4 x \cdot 3 + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.2

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

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Step 1.6.5.1.2.1

Move $x$ .

$\frac{4 \cdot 2 (x \cdot x) + 4 x \cdot 3 + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.2.2

Multiply $x$ by $x$ .

$\frac{4 \cdot 2 x^{2} + 4 x \cdot 3 + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.3

Multiply $4$ by $2$ .

$\frac{8 x^{2} + 4 x \cdot 3 + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.4

Multiply $3$ by $4$ .

$\frac{8 x^{2} + 12 x + 6 (2 x) + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.5

Multiply $2$ by $6$ .

$\frac{8 x^{2} + 12 x + 12 x + 6 \cdot 3 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.1.6

Multiply $6$ by $3$ .

$\frac{8 x^{2} + 12 x + 12 x + 18 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.5.2

Add $12 x$ and $12 x$ .

$\frac{8 x^{2} + 24 x + 18 - 25 (x - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.6

Apply the distributive property.

$\frac{8 x^{2} + 24 x + 18 + (- 25 x - 25 \cdot - 3) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.7

Multiply $- 25$ by $- 3$ .

$\frac{8 x^{2} + 24 x + 18 + (- 25 x + 75) (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.8

Expand $(- 25 x + 75) (x - 3)$ using the FOIL Method.

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Step 1.6.8.1

Apply the distributive property.

$\frac{8 x^{2} + 24 x + 18 - 25 x (x - 3) + 75 (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.8.2

Apply the distributive property.

$\frac{8 x^{2} + 24 x + 18 - 25 x \cdot x - 25 x \cdot - 3 + 75 (x - 3)}{(2 x + 3) (x - 3)} = 5$

Step 1.6.8.3

Apply the distributive property.

$\frac{8 x^{2} + 24 x + 18 - 25 x \cdot x - 25 x \cdot - 3 + 75 x + 75 \cdot - 3}{(2 x + 3) (x - 3)} = 5$

Step 1.6.9

Simplify and combine like terms.

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Step 1.6.9.1

Simplify each term.

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Step 1.6.9.1.1

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

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Step 1.6.9.1.1.1

Move $x$ .

$\frac{8 x^{2} + 24 x + 18 - 25 (x \cdot x) - 25 x \cdot - 3 + 75 x + 75 \cdot - 3}{(2 x + 3) (x - 3)} = 5$

Step 1.6.9.1.1.2

Multiply $x$ by $x$ .

$\frac{8 x^{2} + 24 x + 18 - 25 x^{2} - 25 x \cdot - 3 + 75 x + 75 \cdot - 3}{(2 x + 3) (x - 3)} = 5$

Step 1.6.9.1.2

Multiply $- 3$ by $- 25$ .

$\frac{8 x^{2} + 24 x + 18 - 25 x^{2} + 75 x + 75 x + 75 \cdot - 3}{(2 x + 3) (x - 3)} = 5$

Step 1.6.9.1.3

Multiply $75$ by $- 3$ .

$\frac{8 x^{2} + 24 x + 18 - 25 x^{2} + 75 x + 75 x - 225}{(2 x + 3) (x - 3)} = 5$

Step 1.6.9.2

Add $75 x$ and $75 x$ .

$\frac{8 x^{2} + 24 x + 18 - 25 x^{2} + 150 x - 225}{(2 x + 3) (x - 3)} = 5$

Step 1.6.10

Subtract $25 x^{2}$ from $8 x^{2}$ .

$\frac{- 17 x^{2} + 24 x + 18 + 150 x - 225}{(2 x + 3) (x - 3)} = 5$

Step 1.6.11

Add $24 x$ and $150 x$ .

$\frac{- 17 x^{2} + 174 x + 18 - 225}{(2 x + 3) (x - 3)} = 5$

Step 1.6.12

Subtract $225$ from $18$ .

$\frac{- 17 x^{2} + 174 x - 207}{(2 x + 3) (x - 3)} = 5$

Step 1.7

Simplify with factoring out.

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Step 1.7.1

Factor $- 1$ out of $- 17 x^{2}$ .

$\frac{- (17 x^{2}) + 174 x - 207}{(2 x + 3) (x - 3)} = 5$

Step 1.7.2

Factor $- 1$ out of $174 x$ .

$\frac{- (17 x^{2}) - (- 174 x) - 207}{(2 x + 3) (x - 3)} = 5$

Step 1.7.3

Factor $- 1$ out of $- (17 x^{2}) - (- 174 x)$ .

$\frac{- (17 x^{2} - 174 x) - 207}{(2 x + 3) (x - 3)} = 5$

Step 1.7.4

Rewrite $- 207$ as $- 1 (207)$ .

$\frac{- (17 x^{2} - 174 x) - 1 (207)}{(2 x + 3) (x - 3)} = 5$

Step 1.7.5

Factor $- 1$ out of $- (17 x^{2} - 174 x) - 1 (207)$ .

$\frac{- (17 x^{2} - 174 x + 207)}{(2 x + 3) (x - 3)} = 5$

Step 1.7.6

Simplify the expression.

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Step 1.7.6.1

Rewrite $- (17 x^{2} - 174 x + 207)$ as $- 1 (17 x^{2} - 174 x + 207)$ .

$\frac{- 1 (17 x^{2} - 174 x + 207)}{(2 x + 3) (x - 3)} = 5$

Step 1.7.6.2

Move the negative in front of the fraction.

$- \frac{17 x^{2} - 174 x + 207}{(2 x + 3) (x - 3)} = 5$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(2 x + 3) (x - 3), 1$

Step 2.2

The LCM of one and any expression is the expression.

$(2 x + 3) (x - 3)$

Step 3

Multiply each term in $- \frac{17 x^{2} - 174 x + 207}{(2 x + 3) (x - 3)} = 5$ by $(2 x + 3) (x - 3)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{17 x^{2} - 174 x + 207}{(2 x + 3) (x - 3)} = 5$ by $(2 x + 3) (x - 3)$ .

$- \frac{17 x^{2} - 174 x + 207}{(2 x + 3) (x - 3)} ((2 x + 3) (x - 3)) = 5 ((2 x + 3) (x - 3))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(2 x + 3) (x - 3)$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{17 x^{2} - 174 x + 207}{(2 x + 3) (x - 3)}$ into the numerator.

$\frac{- (17 x^{2} - 174 x + 207)}{(2 x + 3) (x - 3)} ((2 x + 3) (x - 3)) = 5 ((2 x + 3) (x - 3))$

Step 3.2.1.2

Cancel the common factor.

$\frac{- (17 x^{2} - 174 x + 207)}{(2 x + 3) (x - 3)} ((2 x + 3) (x - 3)) = 5 ((2 x + 3) (x - 3))$

Step 3.2.1.3

Rewrite the expression.

$- (17 x^{2} - 174 x + 207) = 5 ((2 x + 3) (x - 3))$

Step 3.2.2

Apply the distributive property.

$- (17 x^{2}) - (- 174 x) - 1 \cdot 207 = 5 ((2 x + 3) (x - 3))$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Multiply $17$ by $- 1$ .

$- 17 x^{2} - (- 174 x) - 1 \cdot 207 = 5 ((2 x + 3) (x - 3))$

Step 3.2.3.2

Multiply $- 174$ by $- 1$ .

$- 17 x^{2} + 174 x - 1 \cdot 207 = 5 ((2 x + 3) (x - 3))$

Step 3.2.3.3

Multiply $- 1$ by $207$ .

$- 17 x^{2} + 174 x - 207 = 5 ((2 x + 3) (x - 3))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Expand $(2 x + 3) (x - 3)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$- 17 x^{2} + 174 x - 207 = 5 (2 x (x - 3) + 3 (x - 3))$

Step 3.3.1.2

Apply the distributive property.

$- 17 x^{2} + 174 x - 207 = 5 (2 x \cdot x + 2 x \cdot - 3 + 3 (x - 3))$

Step 3.3.1.3

Apply the distributive property.

$- 17 x^{2} + 174 x - 207 = 5 (2 x \cdot x + 2 x \cdot - 3 + 3 x + 3 \cdot - 3)$

Step 3.3.2

Simplify and combine like terms.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

Move $x$ .

$- 17 x^{2} + 174 x - 207 = 5 (2 (x \cdot x) + 2 x \cdot - 3 + 3 x + 3 \cdot - 3)$

Step 3.3.2.1.1.2

Multiply $x$ by $x$ .

$- 17 x^{2} + 174 x - 207 = 5 (2 x^{2} + 2 x \cdot - 3 + 3 x + 3 \cdot - 3)$

Step 3.3.2.1.2

Multiply $- 3$ by $2$ .

$- 17 x^{2} + 174 x - 207 = 5 (2 x^{2} - 6 x + 3 x + 3 \cdot - 3)$

Step 3.3.2.1.3

Multiply $3$ by $- 3$ .

$- 17 x^{2} + 174 x - 207 = 5 (2 x^{2} - 6 x + 3 x - 9)$

Step 3.3.2.2

Add $- 6 x$ and $3 x$ .

$- 17 x^{2} + 174 x - 207 = 5 (2 x^{2} - 3 x - 9)$

Step 3.3.3

Apply the distributive property.

$- 17 x^{2} + 174 x - 207 = 5 (2 x^{2}) + 5 (- 3 x) + 5 \cdot - 9$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Multiply $2$ by $5$ .

$- 17 x^{2} + 174 x - 207 = 10 x^{2} + 5 (- 3 x) + 5 \cdot - 9$

Step 3.3.4.2

Multiply $- 3$ by $5$ .

$- 17 x^{2} + 174 x - 207 = 10 x^{2} - 15 x + 5 \cdot - 9$

Step 3.3.4.3

Multiply $5$ by $- 9$ .

$- 17 x^{2} + 174 x - 207 = 10 x^{2} - 15 x - 45$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Subtract $10 x^{2}$ from both sides of the equation.

$- 17 x^{2} + 174 x - 207 - 10 x^{2} = - 15 x - 45$

Step 4.1.2

Add $15 x$ to both sides of the equation.

$- 17 x^{2} + 174 x - 207 - 10 x^{2} + 15 x = - 45$

Step 4.1.3

Subtract $10 x^{2}$ from $- 17 x^{2}$ .

$- 27 x^{2} + 174 x - 207 + 15 x = - 45$

Step 4.1.4

Add $174 x$ and $15 x$ .

$- 27 x^{2} + 189 x - 207 = - 45$

Step 4.2

Add $45$ to both sides of the equation.

$- 27 x^{2} + 189 x - 207 + 45 = 0$

Step 4.3

Add $- 207$ and $45$ .

$- 27 x^{2} + 189 x - 162 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 27$ out of $- 27 x^{2} + 189 x - 162$ .

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Step 4.4.1.1

Factor $- 27$ out of $- 27 x^{2}$ .

$- 27 x^{2} + 189 x - 162 = 0$

Step 4.4.1.2

Factor $- 27$ out of $189 x$ .

$- 27 x^{2} - 27 (- 7 x) - 162 = 0$

Step 4.4.1.3

Factor $- 27$ out of $- 162$ .

$- 27 x^{2} - 27 (- 7 x) - 27 \cdot 6 = 0$

Step 4.4.1.4

Factor $- 27$ out of $- 27 (x^{2}) - 27 (- 7 x)$ .

$- 27 (x^{2} - 7 x) - 27 \cdot 6 = 0$

Step 4.4.1.5

Factor $- 27$ out of $- 27 (x^{2} - 7 x) - 27 (6)$ .

$- 27 (x^{2} - 7 x + 6) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 4.4.2.1.1

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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 4.4.2.1.2

Write the factored form using these integers.

$- 27 ((x - 6) (x - 1)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- 27 (x - 6) (x - 1) = 0$

Step 4.5

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

$x - 6 = 0$

$x - 1 = 0$

Step 4.6

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.6.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4.7

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.7.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.8

The final solution is all the values that make $- 27 (x - 6) (x - 1) = 0$ true.

$x = 6, 1$