Basic Math Examples

$2 k (k + 5) = 5$

Step 1

Divide each term in

$2 k (k + 5) = 5$ by

$2$ and simplify.

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Step 1.1

Divide each term in

$2 k (k + 5) = 5$ by

$2$ .

$\frac{2 k (k + 5)}{2} = \frac{5}{2}$

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.

$\frac{2 k (k + 5)}{2} = \frac{5}{2}$

Step 1.2.1.2

Divide

$k (k + 5)$ by

$1$ .

$k (k + 5) = \frac{5}{2}$

Step 1.2.2

Apply the distributive property.

$k \cdot k + k \cdot 5 = \frac{5}{2}$

Step 1.2.3

Simplify the expression.

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Step 1.2.3.1

Multiply

$k$ by

$k$ .

$k^{2} + k \cdot 5 = \frac{5}{2}$

Step 1.2.3.2

Move

$5$ to the left of

$k$ .

$k^{2} + 5 k = \frac{5}{2}$

Step 2

Subtract

$\frac{5}{2}$ from both sides of the equation.

$k^{2} + 5 k - \frac{5}{2} = 0$

Step 3

Multiply through by the least common denominator

$2$ , then simplify.

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Step 3.1

Apply the distributive property.

$2 k^{2} + 2 (5 k) + 2 (- \frac{5}{2}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply

$5$ by

$2$ .

$2 k^{2} + 10 k + 2 (- \frac{5}{2}) = 0$

Step 3.2.2

Cancel the common factor of

$2$ .

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Step 3.2.2.1

Move the leading negative in

$- \frac{5}{2}$ into the numerator.

$2 k^{2} + 10 k + 2 (\frac{- 5}{2}) = 0$

Step 3.2.2.2

Cancel the common factor.

$2 k^{2} + 10 k + 2 (\frac{- 5}{2}) = 0$

Step 3.2.2.3

Rewrite the expression.

$2 k^{2} + 10 k - 5 = 0$

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

$a = 2$ ,

$b = 10$ , and

$c = - 5$ into the quadratic formula and solve for

$k$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot - 5)}}{2 \cdot 2}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise

$10$ to the power of

$2$ .

$k = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2

Multiply

$- 4 \cdot 2 \cdot - 5$ .

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Step 6.1.2.1

Multiply

$- 4$ by

$2$ .

$k = \frac{- 10 \pm \sqrt{100 - 8 \cdot - 5}}{2 \cdot 2}$

Step 6.1.2.2

Multiply

$- 8$ by

$- 5$ .

$k = \frac{- 10 \pm \sqrt{100 + 40}}{2 \cdot 2}$

Step 6.1.3

Add

$100$ and

$40$ .

$k = \frac{- 10 \pm \sqrt{140}}{2 \cdot 2}$

Step 6.1.4

Rewrite

$140$ as

$2^{2} \cdot 35$ .

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Step 6.1.4.1

Factor

$4$ out of

$140$ .

$k = \frac{- 10 \pm \sqrt{4 (35)}}{2 \cdot 2}$

Step 6.1.4.2

Rewrite

$4$ as

$2^{2}$ .

$k = \frac{- 10 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 2}$

Step 6.1.5

Pull terms out from under the radical.

$k = \frac{- 10 \pm 2 \sqrt{35}}{2 \cdot 2}$

Step 6.2

Multiply

$2$ by

$2$ .

$k = \frac{- 10 \pm 2 \sqrt{35}}{4}$

Step 6.3

Simplify

$\frac{- 10 \pm 2 \sqrt{35}}{4}$ .

$k = \frac{- 5 \pm \sqrt{35}}{2}$

Step 7

The final answer is the combination of both solutions.

$k = - \frac{5 - \sqrt{35}}{2}, - \frac{5 + \sqrt{35}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{5 - \sqrt{35}}{2}, - \frac{5 + \sqrt{35}}{2}$

Decimal Form:

$k = 0.45803989 \dots, - 5.45803989 \dots$