Algebra Examples

$2 (a - 2) = - \frac{a - 1}{a - 4}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$1, a - 4$

Step 1.2

Remove parentheses.

$1, a - 4$

Step 1.3

The LCM of one and any expression is the expression.

$a - 4$

Step 2

Multiply each term in $2 (a - 2) = - \frac{a - 1}{a - 4}$ by $a - 4$ to eliminate the fractions.

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Step 2.1

Multiply each term in $2 (a - 2) = - \frac{a - 1}{a - 4}$ by $a - 4$ .

$2 (a - 2) (a - 4) = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify by multiplying through.

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Step 2.2.1.1

Apply the distributive property.

$(2 a + 2 \cdot - 2) (a - 4) = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.1.2

Multiply $2$ by $- 2$ .

$(2 a - 4) (a - 4) = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.2

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

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Step 2.2.2.1

Apply the distributive property.

$2 a (a - 4) - 4 (a - 4) = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.2.2

Apply the distributive property.

$2 a \cdot a + 2 a \cdot - 4 - 4 (a - 4) = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.2.3

Apply the distributive property.

$2 a \cdot a + 2 a \cdot - 4 - 4 a - 4 \cdot - 4 = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.3.1.1.1

Move $a$ .

$2 (a \cdot a) + 2 a \cdot - 4 - 4 a - 4 \cdot - 4 = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.3.1.1.2

Multiply $a$ by $a$ .

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

Step 2.2.3.1.2

Multiply $- 4$ by $2$ .

$2 a^{2} - 8 a - 4 a - 4 \cdot - 4 = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.3.1.3

Multiply $- 4$ by $- 4$ .

$2 a^{2} - 8 a - 4 a + 16 = - \frac{a - 1}{a - 4} (a - 4)$

Step 2.2.3.2

Subtract $4 a$ from $- 8 a$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $a - 4$ .

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Step 2.3.1.1

Move the leading negative in $- \frac{a - 1}{a - 4}$ into the numerator.

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

Step 2.3.1.2

Cancel the common factor.

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

Step 2.3.1.3

Rewrite the expression.

$2 a^{2} - 12 a + 16 = - (a - 1)$

Step 2.3.2

Apply the distributive property.

$2 a^{2} - 12 a + 16 = - a - - 1$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$2 a^{2} - 12 a + 16 = - a + 1$

Step 3

Solve the equation.

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Step 3.1

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

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Step 3.1.1

Add $a$ to both sides of the equation.

$2 a^{2} - 12 a + 16 + a = 1$

Step 3.1.2

Add $- 12 a$ and $a$ .

$2 a^{2} - 11 a + 16 = 1$

Step 3.2

Subtract $1$ from both sides of the equation.

$2 a^{2} - 11 a + 16 - 1 = 0$

Step 3.3

Subtract $1$ from $16$ .

$2 a^{2} - 11 a + 15 = 0$

Step 3.4

Factor by grouping.

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Step 3.4.1

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 $a \cdot c = 2 \cdot 15 = 30$ and whose sum is $b = - 11$ .

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Step 3.4.1.1

Factor $- 11$ out of $- 11 a$ .

$2 a^{2} - 11 a + 15 = 0$

Step 3.4.1.2

Rewrite $- 11$ as $- 5$ plus $- 6$

$2 a^{2} + (- 5 - 6) a + 15 = 0$

Step 3.4.1.3

Apply the distributive property.

$2 a^{2} - 5 a - 6 a + 15 = 0$

Step 3.4.2

Factor out the greatest common factor from each group.

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Step 3.4.2.1

Group the first two terms and the last two terms.

$(2 a^{2} - 5 a) - 6 a + 15 = 0$

Step 3.4.2.2

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

$a (2 a - 5) - 3 (2 a - 5) = 0$

Step 3.4.3

Factor the polynomial by factoring out the greatest common factor, $2 a - 5$ .

$(2 a - 5) (a - 3) = 0$

Step 3.5

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

$2 a - 5 = 0$

$a - 3 = 0$

Step 3.6

Set $2 a - 5$ equal to $0$ and solve for $a$ .

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Step 3.6.1

Set $2 a - 5$ equal to $0$ .

$2 a - 5 = 0$

Step 3.6.2

Solve $2 a - 5 = 0$ for $a$ .

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Step 3.6.2.1

Add $5$ to both sides of the equation.

$2 a = 5$

Step 3.6.2.2

Divide each term in $2 a = 5$ by $2$ and simplify.

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Step 3.6.2.2.1

Divide each term in $2 a = 5$ by $2$ .

$\frac{2 a}{2} = \frac{5}{2}$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{5}{2}$

Step 3.6.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{5}{2}$

Step 3.7

Set $a - 3$ equal to $0$ and solve for $a$ .

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Step 3.7.1

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Step 3.7.2

Add $3$ to both sides of the equation.

$a = 3$

Step 3.8

The final solution is all the values that make $(2 a - 5) (a - 3) = 0$ true.

$a = \frac{5}{2}, 3$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = \frac{5}{2}, 3$

Decimal Form:

$a = 2.5, 3$

Mixed Number Form:

$a = 2 \frac{1}{2}, 3$