Trigonometry Examples

$9 x^{2} + 3 x - 4 = 0$

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 $a = 9$ , $b = 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 3.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 3.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36 \cdot - 4}}{2 \cdot 9}$

Step 3.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{9 + 144}}{2 \cdot 9}$

Step 3.1.3

Add $9$ and $144$ .

$x = \frac{- 3 \pm \sqrt{153}}{2 \cdot 9}$

Step 3.1.4

Rewrite $153$ as $3^{2} \cdot 17$ .

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Step 3.1.4.1

Factor $9$ out of $153$ .

$x = \frac{- 3 \pm \sqrt{9 (17)}}{2 \cdot 9}$

Step 3.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 9}$

Step 3.1.5

Pull terms out from under the radical.

$x = \frac{- 3 \pm 3 \sqrt{17}}{2 \cdot 9}$

Step 3.2

Multiply $2$ by $9$ .

$x = \frac{- 3 \pm 3 \sqrt{17}}{18}$

Step 3.3

Simplify $\frac{- 3 \pm 3 \sqrt{17}}{18}$ .

$x = \frac{- 1 \pm \sqrt{17}}{6}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{17}}{6}, - \frac{1 + \sqrt{17}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{17}}{6}, - \frac{1 + \sqrt{17}}{6}$

Decimal Form:

$x = 0.52051760 \dots, - 0.85385093 \dots$