Basic Math Examples

36 = (8 w^{2} + 10 w)

$36 = (8 w^{2} + 10 w)$

Step 1

Rewrite the equation as

8 w^{2} + 10 w = 36

$8 w^{2} + 10 w = 36$ .

8 w^{2} + 10 w = 36

$8 w^{2} + 10 w = 36$

Step 2

Subtract

36

$36$ from both sides of the equation.

8 w^{2} + 10 w - 36 = 0

$8 w^{2} + 10 w - 36 = 0$

Step 3

Factor

2

$2$ out of

8 w^{2} + 10 w - 36

$8 w^{2} + 10 w - 36$ .

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Step 3.1

Factor

2

$2$ out of

8 w^{2}

$8 w^{2}$ .

2 (4 w^{2}) + 10 w - 36 = 0

$2 (4 w^{2}) + 10 w - 36 = 0$

Step 3.2

Factor

2

$2$ out of

10 w

$10 w$ .

2 (4 w^{2}) + 2 (5 w) - 36 = 0

$2 (4 w^{2}) + 2 (5 w) - 36 = 0$

Step 3.3

Factor

2

$2$ out of

- 36

$- 36$ .

2 (4 w^{2}) + 2 (5 w) + 2 (- 18) = 0

$2 (4 w^{2}) + 2 (5 w) + 2 (- 18) = 0$

Step 3.4

Factor

2

$2$ out of

2 (4 w^{2}) + 2 (5 w)

$2 (4 w^{2}) + 2 (5 w)$ .

2 (4 w^{2} + 5 w) + 2 (- 18) = 0

$2 (4 w^{2} + 5 w) + 2 (- 18) = 0$

Step 3.5

Factor

2

$2$ out of

2 (4 w^{2} + 5 w) + 2 (- 18)

$2 (4 w^{2} + 5 w) + 2 (- 18)$ .

2 (4 w^{2} + 5 w - 18) = 0

$2 (4 w^{2} + 5 w - 18) = 0$

2 (4 w^{2} + 5 w - 18) = 0

$2 (4 w^{2} + 5 w - 18) = 0$

Step 4

Divide each term in

2 (4 w^{2} + 5 w - 18) = 0

$2 (4 w^{2} + 5 w - 18) = 0$ by

2

$2$ and simplify.

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Step 4.1

Divide each term in

2 (4 w^{2} + 5 w - 18) = 0

$2 (4 w^{2} + 5 w - 18) = 0$ by

2

$2$ .

\frac{2 (4 w^{2} + 5 w - 18)}{2} = \frac{0}{2}

$\frac{2 (4 w^{2} + 5 w - 18)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

2

$2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (4 w^{2} + 5 w - 18)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide

$4 w^{2} + 5 w - 18$ by

$1$ .

$4 w^{2} + 5 w - 18 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide

$0$ by

$2$ .

$4 w^{2} + 5 w - 18 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values

$a = 4$ ,

$b = 5$ , and

$c = - 18$ into the quadratic formula and solve for

$w$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot - 18)}}{2 \cdot 4}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise

$5$ to the power of

$2$ .

$w = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 18}}{2 \cdot 4}$

Step 7.1.2

Multiply

$- 4 \cdot 4 \cdot - 18$ .

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Step 7.1.2.1

Multiply

$- 4$ by

$4$ .

$w = \frac{- 5 \pm \sqrt{25 - 16 \cdot - 18}}{2 \cdot 4}$

Step 7.1.2.2

Multiply

$- 16$ by

$- 18$ .

$w = \frac{- 5 \pm \sqrt{25 + 288}}{2 \cdot 4}$

Step 7.1.3

Add

$25$ and

$288$ .

$w = \frac{- 5 \pm \sqrt{313}}{2 \cdot 4}$

Step 7.2

Multiply

$2$ by

$4$ .

$w = \frac{- 5 \pm \sqrt{313}}{8}$

Step 8

The final answer is the combination of both solutions.

$w = - \frac{5 - \sqrt{313}}{8}, - \frac{5 + \sqrt{313}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{5 - \sqrt{313}}{8}, - \frac{5 + \sqrt{313}}{8}$

Decimal Form:

$w = 1.58647575 \dots, - 2.83647575 \dots$