Finite Math Examples

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

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

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

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 4 \cdot - 3}}{2 \cdot - 4}$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $- 4$ by $- 4$ .

$x = \frac{- 9 \pm \sqrt{81 + 16 \cdot - 3}}{2 \cdot - 4}$

Step 3.1.2.2

Multiply $16$ by $- 3$ .

$x = \frac{- 9 \pm \sqrt{81 - 48}}{2 \cdot - 4}$

Step 3.1.3

Subtract $48$ from $81$ .

$x = \frac{- 9 \pm \sqrt{33}}{2 \cdot - 4}$

Step 3.2

Multiply $2$ by $- 4$ .

$x = \frac{- 9 \pm \sqrt{33}}{- 8}$

Step 3.3

Simplify $\frac{- 9 \pm \sqrt{33}}{- 8}$ .

$x = \frac{9 \pm \sqrt{33}}{8}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{33}}{8}, \frac{9 - \sqrt{33}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{33}}{8}, \frac{9 - \sqrt{33}}{8}$

Decimal Form:

$x = 1.84307033 \dots, 0.40692966 \dots$