Basic Math Examples

$- \frac{4 a}{a + 4} = \frac{12}{a - 11}$

Step 1

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.

$- 4 a (a - 11) = (a + 4) \cdot 12$

Step 2

Solve the equation for $a$ .

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Step 2.1

Simplify $- 4 a (a - 11)$ .

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Step 2.1.1

Rewrite.

$0 + 0 - 4 a (a - 11) = (a + 4) \cdot 12$

Step 2.1.2

Simplify by adding zeros.

$- 4 a (a - 11) = (a + 4) \cdot 12$

Step 2.1.3

Apply the distributive property.

$- 4 a \cdot a - 4 a \cdot - 11 = (a + 4) \cdot 12$

Step 2.1.4

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

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Step 2.1.4.1

Move $a$ .

$- 4 (a \cdot a) - 4 a \cdot - 11 = (a + 4) \cdot 12$

Step 2.1.4.2

Multiply $a$ by $a$ .

$- 4 a^{2} - 4 a \cdot - 11 = (a + 4) \cdot 12$

Step 2.1.5

Multiply $- 11$ by $- 4$ .

$- 4 a^{2} + 44 a = (a + 4) \cdot 12$

Step 2.2

Simplify $(a + 4) \cdot 12$ .

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Step 2.2.1

Apply the distributive property.

$- 4 a^{2} + 44 a = a \cdot 12 + 4 \cdot 12$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

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

$- 4 a^{2} + 44 a = 12 \cdot a + 4 \cdot 12$

Step 2.2.2.2

Multiply $4$ by $12$ .

$- 4 a^{2} + 44 a = 12 a + 48$

Step 2.3

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

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Step 2.3.1

Subtract $12 a$ from both sides of the equation.

$- 4 a^{2} + 44 a - 12 a = 48$

Step 2.3.2

Subtract $12 a$ from $44 a$ .

$- 4 a^{2} + 32 a = 48$

Step 2.4

Subtract $48$ from both sides of the equation.

$- 4 a^{2} + 32 a - 48 = 0$

Step 2.5

Factor the left side of the equation.

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Step 2.5.1

Factor $- 4$ out of $- 4 a^{2} + 32 a - 48$ .

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Step 2.5.1.1

Factor $- 4$ out of $- 4 a^{2}$ .

$- 4 a^{2} + 32 a - 48 = 0$

Step 2.5.1.2

Factor $- 4$ out of $32 a$ .

$- 4 a^{2} - 4 (- 8 a) - 48 = 0$

Step 2.5.1.3

Factor $- 4$ out of $- 48$ .

$- 4 a^{2} - 4 (- 8 a) - 4 \cdot 12 = 0$

Step 2.5.1.4

Factor $- 4$ out of $- 4 (a^{2}) - 4 (- 8 a)$ .

$- 4 (a^{2} - 8 a) - 4 \cdot 12 = 0$

Step 2.5.1.5

Factor $- 4$ out of $- 4 (a^{2} - 8 a) - 4 (12)$ .

$- 4 (a^{2} - 8 a + 12) = 0$

Step 2.5.2

Factor.

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Step 2.5.2.1

Factor $a^{2} - 8 a + 12$ using the AC method.

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Step 2.5.2.1.1

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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 2.5.2.1.2

Write the factored form using these integers.

$- 4 ((a - 6) (a - 2)) = 0$

Step 2.5.2.2

Remove unnecessary parentheses.

$- 4 (a - 6) (a - 2) = 0$

Step 2.6

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

$a - 6 = 0$

$a - 2 = 0$

Step 2.7

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

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Step 2.7.1

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

$a - 6 = 0$

Step 2.7.2

Add $6$ to both sides of the equation.

$a = 6$

Step 2.8

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

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Step 2.8.1

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

$a - 2 = 0$

Step 2.8.2

Add $2$ to both sides of the equation.

$a = 2$

Step 2.9

The final solution is all the values that make $- 4 (a - 6) (a - 2) = 0$ true.

$a = 6, 2$