Finite Math Examples

$2 x^{2} + 3 x - \frac{9}{2}$

Step 1

Set $2 x^{2} + 3 x - \frac{9}{2}$ equal to $0$ .

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

Step 2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.1

Apply the distributive property.

$2 (2 x^{2}) + 2 (3 x) + 2 (- \frac{9}{2}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $2$ by $2$ .

$4 x^{2} + 2 (3 x) + 2 (- \frac{9}{2}) = 0$

Step 2.2.2

Multiply $3$ by $2$ .

$4 x^{2} + 6 x + 2 (- \frac{9}{2}) = 0$

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$4 x^{2} + 6 x + 2 (\frac{- 9}{2}) = 0$

Step 2.2.3.2

Cancel the common factor.

$4 x^{2} + 6 x + 2 (\frac{- 9}{2}) = 0$

Step 2.2.3.3

Rewrite the expression.

$4 x^{2} + 6 x - 9 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 4$ , $b = 6$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (4 \cdot - 9)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 4 \cdot - 9}}{2 \cdot 4}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 - 16 \cdot - 9}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 9$ .

$x = \frac{- 6 \pm \sqrt{36 + 144}}{2 \cdot 4}$

Step 5.1.3

Add $36$ and $144$ .

$x = \frac{- 6 \pm \sqrt{180}}{2 \cdot 4}$

Step 5.1.4

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 5.1.4.1

Factor $36$ out of $180$ .

$x = \frac{- 6 \pm \sqrt{36 (5)}}{2 \cdot 4}$

Step 5.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm \sqrt{6^{2} \cdot 5}}{2 \cdot 4}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 6 \sqrt{5}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 6 \pm 6 \sqrt{5}}{8}$

Step 5.3

Simplify $\frac{- 6 \pm 6 \sqrt{5}}{8}$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.92705098 \dots, - 2.42705098 \dots$