Trigonometry Examples

$4 x^{2} + 5 x + 3 = 0$

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 $a = 4$ , $b = 5$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot 3)}}{2 \cdot 4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot 3}}{2 \cdot 4}$

Step 3.1.2

Multiply $- 4 \cdot 4 \cdot 3$ .

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Step 3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 \cdot 3}}{2 \cdot 4}$

Step 3.1.2.2

Multiply $- 16$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 4}$

Step 3.1.3

Subtract $48$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 4}$

Step 3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 4}$

Step 3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 4}$

Step 3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 4}$

Step 3.2

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm i \sqrt{23}}{8}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{23}}{8}, - \frac{5 + i \sqrt{23}}{8}$