Finite Math Examples

$7 x^{2} - 2 x - 8 = 0$

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 $a = 7$ , $b = - 2$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (7 \cdot - 8)}}{2 \cdot 7}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 7 \cdot - 8}}{2 \cdot 7}$

Step 3.1.2

Multiply $- 4 \cdot 7 \cdot - 8$ .

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Step 3.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{2 \pm \sqrt{4 - 28 \cdot - 8}}{2 \cdot 7}$

Step 3.1.2.2

Multiply $- 28$ by $- 8$ .

$x = \frac{2 \pm \sqrt{4 + 224}}{2 \cdot 7}$

Step 3.1.3

Add $4$ and $224$ .

$x = \frac{2 \pm \sqrt{228}}{2 \cdot 7}$

Step 3.1.4

Rewrite $228$ as $2^{2} \cdot 57$ .

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Step 3.1.4.1

Factor $4$ out of $228$ .

$x = \frac{2 \pm \sqrt{4 (57)}}{2 \cdot 7}$

Step 3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 57}}{2 \cdot 7}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{57}}{2 \cdot 7}$

Step 3.2

Multiply $2$ by $7$ .

$x = \frac{2 \pm 2 \sqrt{57}}{14}$

Step 3.3

Simplify $\frac{2 \pm 2 \sqrt{57}}{14}$ .

$x = \frac{1 \pm \sqrt{57}}{7}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 7 \cdot - 8}}{2 \cdot 7}$

Step 4.1.2

Multiply $- 4 \cdot 7 \cdot - 8$ .

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Step 4.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{2 \pm \sqrt{4 - 28 \cdot - 8}}{2 \cdot 7}$

Step 4.1.2.2

Multiply $- 28$ by $- 8$ .

$x = \frac{2 \pm \sqrt{4 + 224}}{2 \cdot 7}$

Step 4.1.3

Add $4$ and $224$ .

$x = \frac{2 \pm \sqrt{228}}{2 \cdot 7}$

Step 4.1.4

Rewrite $228$ as $2^{2} \cdot 57$ .

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Step 4.1.4.1

Factor $4$ out of $228$ .

$x = \frac{2 \pm \sqrt{4 (57)}}{2 \cdot 7}$

Step 4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 57}}{2 \cdot 7}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{57}}{2 \cdot 7}$

Step 4.2

Multiply $2$ by $7$ .

$x = \frac{2 \pm 2 \sqrt{57}}{14}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{57}}{14}$ .

$x = \frac{1 \pm \sqrt{57}}{7}$

Step 4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{57}}{7}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 7 \cdot - 8}}{2 \cdot 7}$

Step 5.1.2

Multiply $- 4 \cdot 7 \cdot - 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{2 \pm \sqrt{4 - 28 \cdot - 8}}{2 \cdot 7}$

Step 5.1.2.2

Multiply $- 28$ by $- 8$ .

$x = \frac{2 \pm \sqrt{4 + 224}}{2 \cdot 7}$

Step 5.1.3

Add $4$ and $224$ .

$x = \frac{2 \pm \sqrt{228}}{2 \cdot 7}$

Step 5.1.4

Rewrite $228$ as $2^{2} \cdot 57$ .

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Step 5.1.4.1

Factor $4$ out of $228$ .

$x = \frac{2 \pm \sqrt{4 (57)}}{2 \cdot 7}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm \sqrt{2^{2} \cdot 57}}{2 \cdot 7}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{57}}{2 \cdot 7}$

Step 5.2

Multiply $2$ by $7$ .

$x = \frac{2 \pm 2 \sqrt{57}}{14}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{57}}{14}$ .

$x = \frac{1 \pm \sqrt{57}}{7}$

Step 5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{57}}{7}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{57}}{7}, \frac{1 - \sqrt{57}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{57}}{7}, \frac{1 - \sqrt{57}}{7}$

Decimal Form:

$x = 1.22140491 \dots, - 0.93569063 \dots$