Algebra Examples

$2 w^{2} - 15 w + 19 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 2$ , $b = - 15$ , and $c = 19$ into the quadratic formula and solve for $w$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (2 \cdot 19)}}{2 \cdot 2}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 15$ to the power of $2$ .

$w = \frac{15 \pm \sqrt{225 - 4 \cdot 2 \cdot 19}}{2 \cdot 2}$

Step 3.1.2

Multiply $- 4 \cdot 2 \cdot 19$ .

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Step 3.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{15 \pm \sqrt{225 - 8 \cdot 19}}{2 \cdot 2}$

Step 3.1.2.2

Multiply $- 8$ by $19$ .

$w = \frac{15 \pm \sqrt{225 - 152}}{2 \cdot 2}$

Step 3.1.3

Subtract $152$ from $225$ .

$w = \frac{15 \pm \sqrt{73}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$w = \frac{15 \pm \sqrt{73}}{4}$

Step 4

The final answer is the combination of both solutions.

$w = \frac{15 + \sqrt{73}}{4}, \frac{15 - \sqrt{73}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$w = \frac{15 + \sqrt{73}}{4}, \frac{15 - \sqrt{73}}{4}$

Decimal Form:

$w = 5.88600093 \dots, 1.61399906 \dots$