Basic Math Examples

$\frac{k^{2} + 3 k - 4}{k - 4} \cdot \frac{k^{2} - 16}{k^{2} - 1} = a$

Step 1

Simplify $\frac{k^{2} + 3 k - 4}{k - 4} \cdot \frac{k^{2} - 16}{k^{2} - 1}$ .

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Step 1.1

Factor $k^{2} + 3 k - 4$ using the AC method.

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Step 1.1.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

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

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 4$ .

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

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 1$ .

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

Step 1.4

Simplify terms.

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Step 1.4.1

Cancel the common factor of $k - 1$ .

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Step 1.4.1.1

Factor $k - 1$ out of $(k + 1) (k - 1)$ .

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

Step 1.4.1.2

Cancel the common factor.

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

Step 1.4.1.3

Rewrite the expression.

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

Step 1.4.2

Cancel the common factor of $k - 4$ .

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Step 1.4.2.1

Factor $k - 4$ out of $(k + 4) (k - 4)$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

$\frac{(k + 4) (k + 4)}{k + 1} = a$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $k + 4$ to the power of $1$ .

$\frac{{(k + 4)}^{1} (k + 4)}{k + 1} = a$

Step 1.5.2

Raise $k + 4$ to the power of $1$ .

$\frac{{(k + 4)}^{1} {(k + 4)}^{1}}{k + 1} = a$

Step 1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(k + 4)}^{1 + 1}}{k + 1} = a$

Step 1.5.4

Add $1$ and $1$ .

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

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.

$k + 1, 1$

Step 2.2

Remove parentheses.

$k + 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$k + 1$

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $k + 1$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

${(k + 4)}^{2} = a (k + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $a$ by $1$ .

${(k + 4)}^{2} = a k + a$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

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

${(k + 4)}^{2} - a k = a$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Rewrite ${(k + 4)}^{2}$ as $(k + 4) (k + 4)$ .

$(k + 4) (k + 4) - a k = a$

Step 4.1.2.2

Expand $(k + 4) (k + 4)$ using the FOIL Method.

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Step 4.1.2.2.1

Apply the distributive property.

$k (k + 4) + 4 (k + 4) - a k = a$

Step 4.1.2.2.2

Apply the distributive property.

$k \cdot k + k \cdot 4 + 4 (k + 4) - a k = a$

Step 4.1.2.2.3

Apply the distributive property.

$k \cdot k + k \cdot 4 + 4 k + 4 \cdot 4 - a k = a$

Step 4.1.2.3

Simplify and combine like terms.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Multiply $k$ by $k$ .

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

Step 4.1.2.3.1.2

Move $4$ to the left of $k$ .

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

Step 4.1.2.3.1.3

Multiply $4$ by $4$ .

$k^{2} + 4 k + 4 k + 16 - a k = a$

Step 4.1.2.3.2

Add $4 k$ and $4 k$ .

$k^{2} + 8 k + 16 - a k = a$

Step 4.2

Subtract $a$ from both sides of the equation.

$k^{2} + 8 k + 16 - a k - a = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 1$ , $b = 8 - a$ , and $c = 16 - a$ into the quadratic formula and solve for $k$ .

$\frac{- (8 - a) \pm \sqrt{{(8 - a)}^{2} - 4 \cdot (1 \cdot (16 - a))}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the distributive property.

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

Step 4.5.1.2

Multiply $- 1$ by $8$ .

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

Step 4.5.1.3

Multiply $- - a$ .

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Step 4.5.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.5.1.3.2

Multiply $a$ by $1$ .

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

Step 4.5.1.4

Rewrite ${(8 - a)}^{2}$ as $(8 - a) (8 - a)$ .

$k = \frac{- 8 + a \pm \sqrt{(8 - a) (8 - a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.5

Expand $(8 - a) (8 - a)$ using the FOIL Method.

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Step 4.5.1.5.1

Apply the distributive property.

$k = \frac{- 8 + a \pm \sqrt{8 (8 - a) - a (8 - a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.5.2

Apply the distributive property.

$k = \frac{- 8 + a \pm \sqrt{8 \cdot 8 + 8 (- a) - a (8 - a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.5.3

Apply the distributive property.

$k = \frac{- 8 + a \pm \sqrt{8 \cdot 8 + 8 (- a) - a \cdot 8 - a (- a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6

Simplify and combine like terms.

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Step 4.5.1.6.1

Simplify each term.

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Step 4.5.1.6.1.1

Multiply $8$ by $8$ .

$k = \frac{- 8 + a \pm \sqrt{64 + 8 (- a) - a \cdot 8 - a (- a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.2

Multiply $- 1$ by $8$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - a \cdot 8 - a (- a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.3

Multiply $8$ by $- 1$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a - a (- a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a - 1 \cdot (- 1 a \cdot a) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.5

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

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Step 4.5.1.6.1.5.1

Move $a$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a - 1 \cdot (- 1 (a \cdot a)) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.5.2

Multiply $a$ by $a$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a - 1 \cdot (- 1 a^{2}) - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a + 1 a^{2} - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.1.7

Multiply $a^{2}$ by $1$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 8 a - 8 a + a^{2} - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.6.2

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

$k = \frac{- 8 + a \pm \sqrt{64 - 16 a + a^{2} - 4 \cdot 1 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.7

Multiply $- 4$ by $1$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 16 a + a^{2} - 4 \cdot (16 - a)}}{2 \cdot 1}$

Step 4.5.1.8

Apply the distributive property.

$k = \frac{- 8 + a \pm \sqrt{64 - 16 a + a^{2} - 4 \cdot 16 - 4 (- a)}}{2 \cdot 1}$

Step 4.5.1.9

Multiply $- 4$ by $16$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 16 a + a^{2} - 64 - 4 (- a)}}{2 \cdot 1}$

Step 4.5.1.10

Multiply $- 1$ by $- 4$ .

$k = \frac{- 8 + a \pm \sqrt{64 - 16 a + a^{2} - 64 + 4 a}}{2 \cdot 1}$

Step 4.5.1.11

Subtract $64$ from $64$ .

$k = \frac{- 8 + a \pm \sqrt{- 16 a + a^{2} + 0 + 4 a}}{2 \cdot 1}$

Step 4.5.1.12

Add $- 16 a$ and $0$ .

$k = \frac{- 8 + a \pm \sqrt{a^{2} - 16 a + 4 a}}{2 \cdot 1}$

Step 4.5.1.13

Add $- 16 a$ and $4 a$ .

$k = \frac{- 8 + a \pm \sqrt{a^{2} - 12 a}}{2 \cdot 1}$

Step 4.5.1.14

Factor $a$ out of $a^{2} - 12 a$ .

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Step 4.5.1.14.1

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

$k = \frac{- 8 + a \pm \sqrt{a \cdot a - 12 a}}{2 \cdot 1}$

Step 4.5.1.14.2

Factor $a$ out of $- 12 a$ .

$k = \frac{- 8 + a \pm \sqrt{a \cdot a + a \cdot - 12}}{2 \cdot 1}$

Step 4.5.1.14.3

Factor $a$ out of $a \cdot a + a \cdot - 12$ .

$k = \frac{- 8 + a \pm \sqrt{a (a - 12)}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$k = \frac{- 8 + a \pm \sqrt{a (a - 12)}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$k = - \frac{8 - a - \sqrt{a (a - 12)}}{2}$

$k = - \frac{8 - a + \sqrt{a (a - 12)}}{2}$

$k = - \frac{8 - a - \sqrt{a (a - 12)}}{2}$

$k = - \frac{8 - a + \sqrt{a (a - 12)}}{2}$