Finite Math Examples

$4 (k - 2) - 8 = 5 k (4 k - 1)$

Step 1

Subtract $5 k (4 k - 1)$ from both sides of the equation.

$4 (k - 2) - 8 - 5 k (4 k - 1) = 0$

Step 2

Simplify $4 (k - 2) - 8 - 5 k (4 k - 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$4 k + 4 \cdot - 2 - 8 - 5 k (4 k - 1) = 0$

Step 2.1.2

Multiply $4$ by $- 2$ .

$4 k - 8 - 8 - 5 k (4 k - 1) = 0$

Step 2.2

Subtract $8$ from $- 8$ .

$4 k - 16 - 5 k (4 k - 1) = 0$

Step 3

Simplify $4 k - 16 - 5 k (4 k - 1)$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$4 k - 16 - 5 k (4 k) - 5 k \cdot - 1 = 0$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$4 k - 16 - 5 \cdot 4 k \cdot k - 5 k \cdot - 1 = 0$

Step 3.1.3

Multiply $- 1$ by $- 5$ .

$4 k - 16 - 5 \cdot 4 k \cdot k + 5 k = 0$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Multiply $k$ by $k$ by adding the exponents.

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Step 3.1.4.1.1

Move $k$ .

$4 k - 16 - 5 \cdot 4 (k \cdot k) + 5 k = 0$

Step 3.1.4.1.2

Multiply $k$ by $k$ .

$4 k - 16 - 5 \cdot 4 k^{2} + 5 k = 0$

Step 3.1.4.2

Multiply $- 5$ by $4$ .

$4 k - 16 - 20 k^{2} + 5 k = 0$

Step 3.2

Add $4 k$ and $5 k$ .

$9 k - 16 - 20 k^{2} = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 20$ , $b = 9$ , and $c = - 16$ into the quadratic formula and solve for $k$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (- 20 \cdot - 16)}}{2 \cdot - 20}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $9$ to the power of $2$ .

$k = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 20 \cdot - 16}}{2 \cdot - 20}$

Step 6.1.2

Multiply $- 4 \cdot - 20 \cdot - 16$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 20$ .

$k = \frac{- 9 \pm \sqrt{81 + 80 \cdot - 16}}{2 \cdot - 20}$

Step 6.1.2.2

Multiply $80$ by $- 16$ .

$k = \frac{- 9 \pm \sqrt{81 - 1280}}{2 \cdot - 20}$

Step 6.1.3

Subtract $1280$ from $81$ .

$k = \frac{- 9 \pm \sqrt{- 1199}}{2 \cdot - 20}$

Step 6.1.4

Rewrite $- 1199$ as $- 1 (1199)$ .

$k = \frac{- 9 \pm \sqrt{- 1 \cdot 1199}}{2 \cdot - 20}$

Step 6.1.5

Rewrite $\sqrt{- 1 (1199)}$ as $\sqrt{- 1} \cdot \sqrt{1199}$ .

$k = \frac{- 9 \pm \sqrt{- 1} \cdot \sqrt{1199}}{2 \cdot - 20}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{2 \cdot - 20}$

Step 6.2

Multiply $2$ by $- 20$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{- 40}$

Step 6.3

Simplify $\frac{- 9 \pm i \sqrt{1199}}{- 40}$ .

$k = \frac{9 \pm i \sqrt{1199}}{40}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $9$ to the power of $2$ .

$k = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 20 \cdot - 16}}{2 \cdot - 20}$

Step 7.1.2

Multiply $- 4 \cdot - 20 \cdot - 16$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 20$ .

$k = \frac{- 9 \pm \sqrt{81 + 80 \cdot - 16}}{2 \cdot - 20}$

Step 7.1.2.2

Multiply $80$ by $- 16$ .

$k = \frac{- 9 \pm \sqrt{81 - 1280}}{2 \cdot - 20}$

Step 7.1.3

Subtract $1280$ from $81$ .

$k = \frac{- 9 \pm \sqrt{- 1199}}{2 \cdot - 20}$

Step 7.1.4

Rewrite $- 1199$ as $- 1 (1199)$ .

$k = \frac{- 9 \pm \sqrt{- 1 \cdot 1199}}{2 \cdot - 20}$

Step 7.1.5

Rewrite $\sqrt{- 1 (1199)}$ as $\sqrt{- 1} \cdot \sqrt{1199}$ .

$k = \frac{- 9 \pm \sqrt{- 1} \cdot \sqrt{1199}}{2 \cdot - 20}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{2 \cdot - 20}$

Step 7.2

Multiply $2$ by $- 20$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{- 40}$

Step 7.3

Simplify $\frac{- 9 \pm i \sqrt{1199}}{- 40}$ .

$k = \frac{9 \pm i \sqrt{1199}}{40}$

Step 7.4

Change the $\pm$ to $+$ .

$k = \frac{9 + i \sqrt{1199}}{40}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $9$ to the power of $2$ .

$k = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 20 \cdot - 16}}{2 \cdot - 20}$

Step 8.1.2

Multiply $- 4 \cdot - 20 \cdot - 16$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 20$ .

$k = \frac{- 9 \pm \sqrt{81 + 80 \cdot - 16}}{2 \cdot - 20}$

Step 8.1.2.2

Multiply $80$ by $- 16$ .

$k = \frac{- 9 \pm \sqrt{81 - 1280}}{2 \cdot - 20}$

Step 8.1.3

Subtract $1280$ from $81$ .

$k = \frac{- 9 \pm \sqrt{- 1199}}{2 \cdot - 20}$

Step 8.1.4

Rewrite $- 1199$ as $- 1 (1199)$ .

$k = \frac{- 9 \pm \sqrt{- 1 \cdot 1199}}{2 \cdot - 20}$

Step 8.1.5

Rewrite $\sqrt{- 1 (1199)}$ as $\sqrt{- 1} \cdot \sqrt{1199}$ .

$k = \frac{- 9 \pm \sqrt{- 1} \cdot \sqrt{1199}}{2 \cdot - 20}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{2 \cdot - 20}$

Step 8.2

Multiply $2$ by $- 20$ .

$k = \frac{- 9 \pm i \sqrt{1199}}{- 40}$

Step 8.3

Simplify $\frac{- 9 \pm i \sqrt{1199}}{- 40}$ .

$k = \frac{9 \pm i \sqrt{1199}}{40}$

Step 8.4

Change the $\pm$ to $-$ .

$k = \frac{9 - i \sqrt{1199}}{40}$

Step 9

The final answer is the combination of both solutions.

$k = \frac{9 + i \sqrt{1199}}{40}, \frac{9 - i \sqrt{1199}}{40}$