Pre-Algebra Examples

$2 k (k + 15) = 15$

Step 1

Divide each term in $2 k (k + 15) = 15$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 k (k + 15) = 15$ by $2$ .

$\frac{2 k (k + 15)}{2} = \frac{15}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 k (k + 15)}{2} = \frac{15}{2}$

Step 1.2.1.2

Divide $k (k + 15)$ by $1$ .

$k (k + 15) = \frac{15}{2}$

Step 1.2.2

Apply the distributive property.

$k \cdot k + k \cdot 15 = \frac{15}{2}$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply $k$ by $k$ .

$k^{2} + k \cdot 15 = \frac{15}{2}$

Step 1.2.3.2

Move $15$ to the left of $k$ .

$k^{2} + 15 k = \frac{15}{2}$

Step 2

Subtract $\frac{15}{2}$ from both sides of the equation.

$k^{2} + 15 k - \frac{15}{2} = 0$

Step 3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.1

Apply the distributive property.

$2 k^{2} + 2 (15 k) + 2 (- \frac{15}{2}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $15$ by $2$ .

$2 k^{2} + 30 k + 2 (- \frac{15}{2}) = 0$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Move the leading negative in $- \frac{15}{2}$ into the numerator.

$2 k^{2} + 30 k + 2 (\frac{- 15}{2}) = 0$

Step 3.2.2.2

Cancel the common factor.

$2 k^{2} + 30 k + 2 (\frac{- 15}{2}) = 0$

Step 3.2.2.3

Rewrite the expression.

$2 k^{2} + 30 k - 15 = 0$

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 $a = 2$ , $b = 30$ , and $c = - 15$ into the quadratic formula and solve for $k$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (2 \cdot - 15)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $30$ to the power of $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 15}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 15$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 8 \cdot - 15}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 15$ .

$k = \frac{- 30 \pm \sqrt{900 + 120}}{2 \cdot 2}$

Step 6.1.3

Add $900$ and $120$ .

$k = \frac{- 30 \pm \sqrt{1020}}{2 \cdot 2}$

Step 6.1.4

Rewrite $1020$ as $2^{2} \cdot 255$ .

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Step 6.1.4.1

Factor $4$ out of $1020$ .

$k = \frac{- 30 \pm \sqrt{4 (255)}}{2 \cdot 2}$

Step 6.1.4.2

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

$k = \frac{- 30 \pm \sqrt{2^{2} \cdot 255}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$k = \frac{- 30 \pm 2 \sqrt{255}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$k = \frac{- 30 \pm 2 \sqrt{255}}{4}$

Step 6.3

Simplify $\frac{- 30 \pm 2 \sqrt{255}}{4}$ .

$k = \frac{- 15 \pm \sqrt{255}}{2}$

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 $30$ to the power of $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 15}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 15$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 8 \cdot - 15}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 15$ .

$k = \frac{- 30 \pm \sqrt{900 + 120}}{2 \cdot 2}$

Step 7.1.3

Add $900$ and $120$ .

$k = \frac{- 30 \pm \sqrt{1020}}{2 \cdot 2}$

Step 7.1.4

Rewrite $1020$ as $2^{2} \cdot 255$ .

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Step 7.1.4.1

Factor $4$ out of $1020$ .

$k = \frac{- 30 \pm \sqrt{4 (255)}}{2 \cdot 2}$

Step 7.1.4.2

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

$k = \frac{- 30 \pm \sqrt{2^{2} \cdot 255}}{2 \cdot 2}$

Step 7.1.5

Pull terms out from under the radical.

$k = \frac{- 30 \pm 2 \sqrt{255}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$k = \frac{- 30 \pm 2 \sqrt{255}}{4}$

Step 7.3

Simplify $\frac{- 30 \pm 2 \sqrt{255}}{4}$ .

$k = \frac{- 15 \pm \sqrt{255}}{2}$

Step 7.4

Change the $\pm$ to $+$ .

$k = \frac{- 15 + \sqrt{255}}{2}$

Step 7.5

Rewrite $- 15$ as $- 1 (15)$ .

$k = \frac{- 1 \cdot 15 + \sqrt{255}}{2}$

Step 7.6

Factor $- 1$ out of $\sqrt{255}$ .

$k = \frac{- 1 \cdot 15 - 1 (- \sqrt{255})}{2}$

Step 7.7

Factor $- 1$ out of $- 1 (15) - 1 (- \sqrt{255})$ .

$k = \frac{- 1 (15 - \sqrt{255})}{2}$

Step 7.8

Move the negative in front of the fraction.

$k = - \frac{15 - \sqrt{255}}{2}$

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 $30$ to the power of $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 4 \cdot 2 \cdot - 15}}{2 \cdot 2}$

Step 8.1.2

Multiply $- 4 \cdot 2 \cdot - 15$ .

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Step 8.1.2.1

Multiply $- 4$ by $2$ .

$k = \frac{- 30 \pm \sqrt{900 - 8 \cdot - 15}}{2 \cdot 2}$

Step 8.1.2.2

Multiply $- 8$ by $- 15$ .

$k = \frac{- 30 \pm \sqrt{900 + 120}}{2 \cdot 2}$

Step 8.1.3

Add $900$ and $120$ .

$k = \frac{- 30 \pm \sqrt{1020}}{2 \cdot 2}$

Step 8.1.4

Rewrite $1020$ as $2^{2} \cdot 255$ .

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Step 8.1.4.1

Factor $4$ out of $1020$ .

$k = \frac{- 30 \pm \sqrt{4 (255)}}{2 \cdot 2}$

Step 8.1.4.2

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

$k = \frac{- 30 \pm \sqrt{2^{2} \cdot 255}}{2 \cdot 2}$

Step 8.1.5

Pull terms out from under the radical.

$k = \frac{- 30 \pm 2 \sqrt{255}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$k = \frac{- 30 \pm 2 \sqrt{255}}{4}$

Step 8.3

Simplify $\frac{- 30 \pm 2 \sqrt{255}}{4}$ .

$k = \frac{- 15 \pm \sqrt{255}}{2}$

Step 8.4

Change the $\pm$ to $-$ .

$k = \frac{- 15 - \sqrt{255}}{2}$

Step 8.5

Rewrite $- 15$ as $- 1 (15)$ .

$k = \frac{- 1 \cdot 15 - \sqrt{255}}{2}$

Step 8.6

Factor $- 1$ out of $- \sqrt{255}$ .

$k = \frac{- 1 \cdot 15 - (\sqrt{255})}{2}$

Step 8.7

Factor $- 1$ out of $- 1 (15) - (\sqrt{255})$ .

$k = \frac{- 1 (15 + \sqrt{255})}{2}$

Step 8.8

Move the negative in front of the fraction.

$k = - \frac{15 + \sqrt{255}}{2}$

Step 9

The final answer is the combination of both solutions.

$k = - \frac{15 - \sqrt{255}}{2}, - \frac{15 + \sqrt{255}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{15 - \sqrt{255}}{2}, - \frac{15 + \sqrt{255}}{2}$

Decimal Form:

$k = 0.48435971 \dots, - 15.48435971 \dots$