Finite Math Examples

$4 w^{2} + 12 w - 44 = 0$

Step 1

Factor $4$ out of $4 w^{2} + 12 w - 44$ .

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Step 1.1

Factor $4$ out of $4 w^{2}$ .

$4 (w^{2}) + 12 w - 44 = 0$

Step 1.2

Factor $4$ out of $12 w$ .

$4 (w^{2}) + 4 (3 w) - 44 = 0$

Step 1.3

Factor $4$ out of $- 44$ .

$4 w^{2} + 4 (3 w) + 4 \cdot - 11 = 0$

Step 1.4

Factor $4$ out of $4 w^{2} + 4 (3 w)$ .

$4 (w^{2} + 3 w) + 4 \cdot - 11 = 0$

Step 1.5

Factor $4$ out of $4 (w^{2} + 3 w) + 4 \cdot - 11$ .

$4 (w^{2} + 3 w - 11) = 0$

Step 2

Divide each term in $4 (w^{2} + 3 w - 11) = 0$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 (w^{2} + 3 w - 11) = 0$ by $4$ .

$\frac{4 (w^{2} + 3 w - 11)}{4} = \frac{0}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 (w^{2} + 3 w - 11)}{4} = \frac{0}{4}$

Step 2.2.1.2

Divide $w^{2} + 3 w - 11$ by $1$ .

$w^{2} + 3 w - 11 = \frac{0}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $4$ .

$w^{2} + 3 w - 11 = 0$

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 $a = 1$ , $b = 3$ , and $c = - 11$ into the quadratic formula and solve for $w$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 11)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $3$ to the power of $2$ .

$w = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 11}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$w = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 11}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 11$ .

$w = \frac{- 3 \pm \sqrt{9 + 44}}{2 \cdot 1}$

Step 5.1.3

Add $9$ and $44$ .

$w = \frac{- 3 \pm \sqrt{53}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$w = \frac{- 3 \pm \sqrt{53}}{2}$

Step 6

The final answer is the combination of both solutions.

$w = - \frac{3 - \sqrt{53}}{2}, - \frac{3 + \sqrt{53}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{3 - \sqrt{53}}{2}, - \frac{3 + \sqrt{53}}{2}$

Decimal Form:

$w = 2.14005494 \dots, - 5.14005494 \dots$