Algebra Examples

$12 (x - a) (x - b) = 12 x^{2} - 7 x - 12$

Step 1

Divide each term in $12 (x - a) (x - b) = 12 x^{2} - 7 x - 12$ by $12 (x - b)$ and simplify.

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Step 1.1

Divide each term in $12 (x - a) (x - b) = 12 x^{2} - 7 x - 12$ by $12 (x - b)$ .

$\frac{12 (x - a) (x - b)}{12 (x - b)} = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $12$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{12 (x - a) (x - b)}{12 (x - b)} = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.2.1.2

Rewrite the expression.

$\frac{(x - a) (x - b)}{x - b} = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.2.2

Cancel the common factor of $x - b$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{(x - a) (x - b)}{x - b} = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.2.2.2

Divide $x - a$ by $1$ .

$x - a = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Cancel the common factor of $12$ .

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Step 1.3.1.1.1

Cancel the common factor.

$x - a = \frac{12 x^{2}}{12 (x - b)} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.3.1.1.2

Rewrite the expression.

$x - a = \frac{x^{2}}{x - b} + \frac{- 7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.3.1.2

Move the negative in front of the fraction.

$x - a = \frac{x^{2}}{x - b} - \frac{7 x}{12 (x - b)} + \frac{- 12}{12 (x - b)}$

Step 1.3.1.3

Cancel the common factor of $- 12$ and $12$ .

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Step 1.3.1.3.1

Factor $12$ out of $- 12$ .

$x - a = \frac{x^{2}}{x - b} - \frac{7 x}{12 (x - b)} + \frac{12 \cdot - 1}{12 (x - b)}$

Step 1.3.1.3.2

Cancel the common factors.

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Step 1.3.1.3.2.1

Cancel the common factor.

$x - a = \frac{x^{2}}{x - b} - \frac{7 x}{12 (x - b)} + \frac{12 \cdot - 1}{12 (x - b)}$

Step 1.3.1.3.2.2

Rewrite the expression.

$x - a = \frac{x^{2}}{x - b} - \frac{7 x}{12 (x - b)} + \frac{- 1}{x - b}$

Step 1.3.1.4

Move the negative in front of the fraction.

$x - a = \frac{x^{2}}{x - b} - \frac{7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.2

To write $\frac{x^{2}}{x - b}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x - a = \frac{x^{2}}{x - b} \cdot \frac{12}{12} - \frac{7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.3

Write each expression with a common denominator of $(x - b) \cdot 12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.3.1

Multiply $\frac{x^{2}}{x - b}$ by $\frac{12}{12}$ .

$x - a = \frac{x^{2} \cdot 12}{(x - b) \cdot 12} - \frac{7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.3.2

Reorder the factors of $(x - b) \cdot 12$ .

$x - a = \frac{x^{2} \cdot 12}{12 (x - b)} - \frac{7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.4

Combine the numerators over the common denominator.

$x - a = \frac{x^{2} \cdot 12 - 7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Factor $x$ out of $x^{2} \cdot 12 - 7 x$ .

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Step 1.3.5.1.1

Factor $x$ out of $x^{2} \cdot 12$ .

$x - a = \frac{x (x \cdot 12) - 7 x}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.5.1.2

Factor $x$ out of $- 7 x$ .

$x - a = \frac{x (x \cdot 12) + x \cdot - 7}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.5.1.3

Factor $x$ out of $x (x \cdot 12) + x \cdot - 7$ .

$x - a = \frac{x (x \cdot 12 - 7)}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.5.2

Move $12$ to the left of $x$ .

$x - a = \frac{x (12 x - 7)}{12 (x - b)} - \frac{1}{x - b}$

Step 1.3.6

To write $- \frac{1}{x - b}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$x - a = \frac{x (12 x - 7)}{12 (x - b)} - \frac{1}{x - b} \cdot \frac{12}{12}$

Step 1.3.7

Write each expression with a common denominator of $12 (x - b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.7.1

Multiply $\frac{1}{x - b}$ by $\frac{12}{12}$ .

$x - a = \frac{x (12 x - 7)}{12 (x - b)} - \frac{12}{(x - b) \cdot 12}$

Step 1.3.7.2

Reorder the factors of $(x - b) \cdot 12$ .

$x - a = \frac{x (12 x - 7)}{12 (x - b)} - \frac{12}{12 (x - b)}$

Step 1.3.8

Combine the numerators over the common denominator.

$x - a = \frac{x (12 x - 7) - 12}{12 (x - b)}$

Step 1.3.9

Simplify the numerator.

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Step 1.3.9.1

Apply the distributive property.

$x - a = \frac{x (12 x) + x \cdot - 7 - 12}{12 (x - b)}$

Step 1.3.9.2

Rewrite using the commutative property of multiplication.

$x - a = \frac{12 x \cdot x + x \cdot - 7 - 12}{12 (x - b)}$

Step 1.3.9.3

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

$x - a = \frac{12 x \cdot x - 7 \cdot x - 12}{12 (x - b)}$

Step 1.3.9.4

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

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Step 1.3.9.4.1

Move $x$ .

$x - a = \frac{12 (x \cdot x) - 7 \cdot x - 12}{12 (x - b)}$

Step 1.3.9.4.2

Multiply $x$ by $x$ .

$x - a = \frac{12 x^{2} - 7 \cdot x - 12}{12 (x - b)}$

$x - a = \frac{12 x^{2} - 7 x - 12}{12 (x - b)}$

Step 1.3.9.5

Factor by grouping.

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Step 1.3.9.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 12 = - 144$ and whose sum is $b = - 7$ .

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Step 1.3.9.5.1.1

Factor $- 7$ out of $- 7 x$ .

$x - a = \frac{12 x^{2} - 7 (x) - 12}{12 (x - b)}$

Step 1.3.9.5.1.2

Rewrite $- 7$ as $9$ plus $- 16$

$x - a = \frac{12 x^{2} + (9 - 16) x - 12}{12 (x - b)}$

Step 1.3.9.5.1.3

Apply the distributive property.

$x - a = \frac{12 x^{2} + 9 x - 16 x - 12}{12 (x - b)}$

Step 1.3.9.5.2

Factor out the greatest common factor from each group.

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Step 1.3.9.5.2.1

Group the first two terms and the last two terms.

$x - a = \frac{(12 x^{2} + 9 x) - 16 x - 12}{12 (x - b)}$

Step 1.3.9.5.2.2

Factor out the greatest common factor (GCF) from each group.

$x - a = \frac{3 x (4 x + 3) - 4 (4 x + 3)}{12 (x - b)}$

Step 1.3.9.5.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$x - a = \frac{(4 x + 3) (3 x - 4)}{12 (x - b)}$

Step 2

Subtract $x$ from both sides of the equation.

$- a = \frac{(4 x + 3) (3 x - 4)}{12 (x - b)} - x$

Step 3

Divide each term in $- a = \frac{(4 x + 3) (3 x - 4)}{12 (x - b)} - x$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- a = \frac{(4 x + 3) (3 x - 4)}{12 (x - b)} - x$ by $- 1$ .

$\frac{- a}{- 1} = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)}}{- 1} + \frac{- x}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)}}{- 1} + \frac{- x}{- 1}$

Step 3.2.2

Divide $a$ by $1$ .

$a = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)}}{- 1} + \frac{- x}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$a = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)} - x}{- 1}$

Step 3.3.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{12 (x - b)}{12 (x - b)}$ .

$a = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)} - x \cdot \frac{12 (x - b)}{12 (x - b)}}{- 1}$

Step 3.3.3

Simplify terms.

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Step 3.3.3.1

Combine $- x$ and $\frac{12 (x - b)}{12 (x - b)}$ .

$a = \frac{\frac{(4 x + 3) (3 x - 4)}{12 (x - b)} + \frac{- x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.3.2

Combine the numerators over the common denominator.

$a = \frac{\frac{(4 x + 3) (3 x - 4) - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

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

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Step 3.3.4.1.1

Apply the distributive property.

$a = \frac{\frac{4 x (3 x - 4) + 3 (3 x - 4) - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.1.2

Apply the distributive property.

$a = \frac{\frac{4 x (3 x) + 4 x \cdot - 4 + 3 (3 x - 4) - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.1.3

Apply the distributive property.

$a = \frac{\frac{4 x (3 x) + 4 x \cdot - 4 + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2

Simplify and combine like terms.

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Step 3.3.4.2.1

Simplify each term.

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Step 3.3.4.2.1.1

Rewrite using the commutative property of multiplication.

$a = \frac{\frac{4 \cdot 3 x \cdot x + 4 x \cdot - 4 + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.2

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

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Step 3.3.4.2.1.2.1

Move $x$ .

$a = \frac{\frac{4 \cdot 3 (x \cdot x) + 4 x \cdot - 4 + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.2.2

Multiply $x$ by $x$ .

$a = \frac{\frac{4 \cdot 3 x^{2} + 4 x \cdot - 4 + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.3

Multiply $4$ by $3$ .

$a = \frac{\frac{12 x^{2} + 4 x \cdot - 4 + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.4

Multiply $- 4$ by $4$ .

$a = \frac{\frac{12 x^{2} - 16 x + 3 (3 x) + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.5

Multiply $3$ by $3$ .

$a = \frac{\frac{12 x^{2} - 16 x + 9 x + 3 \cdot - 4 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.1.6

Multiply $3$ by $- 4$ .

$a = \frac{\frac{12 x^{2} - 16 x + 9 x - 12 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.2.2

Add $- 16 x$ and $9 x$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - x (12 (x - b))}{12 (x - b)}}{- 1}$

Step 3.3.4.3

Apply the distributive property.

$a = \frac{\frac{12 x^{2} - 7 x - 12 - x (12 x + 12 (- b))}{12 (x - b)}}{- 1}$

Step 3.3.4.4

Multiply $- 1$ by $12$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - x (12 x - 12 b)}{12 (x - b)}}{- 1}$

Step 3.3.4.5

Apply the distributive property.

$a = \frac{\frac{12 x^{2} - 7 x - 12 - x (12 x) - x (- 12 b)}{12 (x - b)}}{- 1}$

Step 3.3.4.6

Rewrite using the commutative property of multiplication.

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 1 \cdot 12 x \cdot x - x (- 12 b)}{12 (x - b)}}{- 1}$

Step 3.3.4.7

Rewrite using the commutative property of multiplication.

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 1 \cdot 12 x \cdot x - 1 \cdot - 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.8

Simplify each term.

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Step 3.3.4.8.1

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

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Step 3.3.4.8.1.1

Move $x$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 1 \cdot 12 (x \cdot x) - 1 \cdot - 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.8.1.2

Multiply $x$ by $x$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 1 \cdot 12 x^{2} - 1 \cdot - 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.8.2

Multiply $- 1$ by $12$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 12 x^{2} - 1 \cdot - 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.8.3

Multiply $- 1$ by $- 12$ .

$a = \frac{\frac{12 x^{2} - 7 x - 12 - 12 x^{2} + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.9

Subtract $12 x^{2}$ from $12 x^{2}$ .

$a = \frac{\frac{- 7 x - 12 + 0 + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.4.10

Add $- 7 x$ and $0$ .

$a = \frac{\frac{- 7 x - 12 + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.5

Simplify terms.

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Step 3.3.5.1

Factor $- 1$ out of $- 7 x$ .

$a = \frac{\frac{- (7 x) - 12 + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.5.2

Rewrite $- 12$ as $- 1 (12)$ .

$a = \frac{\frac{- (7 x) - 1 (12) + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.5.3

Factor $- 1$ out of $- (7 x) - 1 (12)$ .

$a = \frac{\frac{- (7 x + 12) + 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.5.4

Factor $- 1$ out of $12 x b$ .

$a = \frac{\frac{- (7 x + 12) - (- 12 x b)}{12 (x - b)}}{- 1}$

Step 3.3.5.5

Factor $- 1$ out of $- (7 x + 12) - (- 12 x b)$ .

$a = \frac{\frac{- (7 x + 12 - 12 x b)}{12 (x - b)}}{- 1}$

Step 3.3.5.6

Simplify the expression.

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Step 3.3.5.6.1

Rewrite $- (7 x + 12 - 12 x b)$ as $- 1 (7 x + 12 - 12 x b)$ .

$a = \frac{\frac{- 1 (7 x + 12 - 12 x b)}{12 (x - b)}}{- 1}$

Step 3.3.5.6.2

Move the negative in front of the fraction.

$a = \frac{- \frac{7 x + 12 - 12 x b}{12 (x - b)}}{- 1}$

Step 3.3.5.7

Dividing two negative values results in a positive value.

$a = \frac{\frac{7 x + 12 - 12 x b}{12 (x - b)}}{1}$

Step 3.3.5.8

Divide $\frac{7 x + 12 - 12 x b}{12 (x - b)}$ by $1$ .

$a = \frac{7 x + 12 - 12 x b}{12 (x - b)}$