Trigonometry Examples

$7 x^{2} - 13 x + 2 = 0$

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 $a = 7$ , $b = - 13$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (7 \cdot 2)}}{2 \cdot 7}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 7 \cdot 2}}{2 \cdot 7}$

Step 3.1.2

Multiply $- 4 \cdot 7 \cdot 2$ .

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Step 3.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{13 \pm \sqrt{169 - 28 \cdot 2}}{2 \cdot 7}$

Step 3.1.2.2

Multiply $- 28$ by $2$ .

$x = \frac{13 \pm \sqrt{169 - 56}}{2 \cdot 7}$

Step 3.1.3

Subtract $56$ from $169$ .

$x = \frac{13 \pm \sqrt{113}}{2 \cdot 7}$

Step 3.2

Multiply $2$ by $7$ .

$x = \frac{13 \pm \sqrt{113}}{14}$

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 $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 7 \cdot 2}}{2 \cdot 7}$

Step 4.1.2

Multiply $- 4 \cdot 7 \cdot 2$ .

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Step 4.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{13 \pm \sqrt{169 - 28 \cdot 2}}{2 \cdot 7}$

Step 4.1.2.2

Multiply $- 28$ by $2$ .

$x = \frac{13 \pm \sqrt{169 - 56}}{2 \cdot 7}$

Step 4.1.3

Subtract $56$ from $169$ .

$x = \frac{13 \pm \sqrt{113}}{2 \cdot 7}$

Step 4.2

Multiply $2$ by $7$ .

$x = \frac{13 \pm \sqrt{113}}{14}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{13 + \sqrt{113}}{14}$

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 $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 7 \cdot 2}}{2 \cdot 7}$

Step 5.1.2

Multiply $- 4 \cdot 7 \cdot 2$ .

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Step 5.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{13 \pm \sqrt{169 - 28 \cdot 2}}{2 \cdot 7}$

Step 5.1.2.2

Multiply $- 28$ by $2$ .

$x = \frac{13 \pm \sqrt{169 - 56}}{2 \cdot 7}$

Step 5.1.3

Subtract $56$ from $169$ .

$x = \frac{13 \pm \sqrt{113}}{2 \cdot 7}$

Step 5.2

Multiply $2$ by $7$ .

$x = \frac{13 \pm \sqrt{113}}{14}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{13 - \sqrt{113}}{14}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{13 + \sqrt{113}}{14}, \frac{13 - \sqrt{113}}{14}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13 + \sqrt{113}}{14}, \frac{13 - \sqrt{113}}{14}$

Decimal Form:

$x = 1.68786755 \dots, 0.16927529 \dots$