Finite Math Examples

$2 w^{2} + 25 w + 50 = 289$

Step 1

Subtract $289$ from both sides of the equation.

$2 w^{2} + 25 w + 50 - 289 = 0$

Step 2

Subtract $289$ from $50$ .

$2 w^{2} + 25 w - 239 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = 25$ , and $c = - 239$ into the quadratic formula and solve for $w$ .

$\frac{- 25 \pm \sqrt{25^{2} - 4 \cdot (2 \cdot - 239)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $25$ to the power of $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 4 \cdot 2 \cdot - 239}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 239$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 8 \cdot - 239}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 239$ .

$w = \frac{- 25 \pm \sqrt{625 + 1912}}{2 \cdot 2}$

Step 5.1.3

Add $625$ and $1912$ .

$w = \frac{- 25 \pm \sqrt{2537}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$w = \frac{- 25 \pm \sqrt{2537}}{4}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $25$ to the power of $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 4 \cdot 2 \cdot - 239}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 239$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 8 \cdot - 239}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 239$ .

$w = \frac{- 25 \pm \sqrt{625 + 1912}}{2 \cdot 2}$

Step 6.1.3

Add $625$ and $1912$ .

$w = \frac{- 25 \pm \sqrt{2537}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$w = \frac{- 25 \pm \sqrt{2537}}{4}$

Step 6.3

Change the $\pm$ to $+$ .

$w = \frac{- 25 + \sqrt{2537}}{4}$

Step 6.4

Rewrite $- 25$ as $- 1 (25)$ .

$w = \frac{- 1 \cdot 25 + \sqrt{2537}}{4}$

Step 6.5

Factor $- 1$ out of $\sqrt{2537}$ .

$w = \frac{- 1 \cdot 25 - 1 (- \sqrt{2537})}{4}$

Step 6.6

Factor $- 1$ out of $- 1 (25) - 1 (- \sqrt{2537})$ .

$w = \frac{- 1 (25 - \sqrt{2537})}{4}$

Step 6.7

Move the negative in front of the fraction.

$w = - \frac{25 - \sqrt{2537}}{4}$

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 $25$ to the power of $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 4 \cdot 2 \cdot - 239}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 239$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{- 25 \pm \sqrt{625 - 8 \cdot - 239}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 239$ .

$w = \frac{- 25 \pm \sqrt{625 + 1912}}{2 \cdot 2}$

Step 7.1.3

Add $625$ and $1912$ .

$w = \frac{- 25 \pm \sqrt{2537}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$w = \frac{- 25 \pm \sqrt{2537}}{4}$

Step 7.3

Change the $\pm$ to $-$ .

$w = \frac{- 25 - \sqrt{2537}}{4}$

Step 7.4

Rewrite $- 25$ as $- 1 (25)$ .

$w = \frac{- 1 \cdot 25 - \sqrt{2537}}{4}$

Step 7.5

Factor $- 1$ out of $- \sqrt{2537}$ .

$w = \frac{- 1 \cdot 25 - (\sqrt{2537})}{4}$

Step 7.6

Factor $- 1$ out of $- 1 (25) - (\sqrt{2537})$ .

$w = \frac{- 1 (25 + \sqrt{2537})}{4}$

Step 7.7

Move the negative in front of the fraction.

$w = - \frac{25 + \sqrt{2537}}{4}$

Step 8

The final answer is the combination of both solutions.

$w = - \frac{25 - \sqrt{2537}}{4}, - \frac{25 + \sqrt{2537}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{25 - \sqrt{2537}}{4}, - \frac{25 + \sqrt{2537}}{4}$

Decimal Form:

$w = 6.34216025 \dots, - 18.84216025 \dots$