Finite Math Examples

$f (x) = \frac{1}{2} x^{2} + 2 x - 10$

Step 1

Set $\frac{1}{2} x^{2} + 2 x - 10$ equal to $0$ .

$\frac{1}{2} x^{2} + 2 x - 10 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} + 2 x - 10 = 0$

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (2 x) + 2 \cdot - 10 = 0$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (2 x) + 2 \cdot - 10 = 0$

Step 2.2.2.1.2

Rewrite the expression.

$x^{2} + 2 (2 x) + 2 \cdot - 10 = 0$

Step 2.2.2.2

Multiply $2$ by $2$ .

$x^{2} + 4 x + 2 \cdot - 10 = 0$

Step 2.2.2.3

Multiply $2$ by $- 10$ .

$x^{2} + 4 x - 20 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 1$ , $b = 4$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 20}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{- 4 \pm \sqrt{16 + 80}}{2 \cdot 1}$

Step 2.5.1.3

Add $16$ and $80$ .

$x = \frac{- 4 \pm \sqrt{96}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 2.5.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 4 \pm \sqrt{16 (6)}}{2 \cdot 1}$

Step 2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 1}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{6}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

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

Step 2.5.3

Simplify $\frac{- 4 \pm 4 \sqrt{6}}{2}$ .

$x = - 2 \pm 2 \sqrt{6}$

Step 2.6

The final answer is the combination of both solutions.

$x = - 2 + 2 \sqrt{6}, - 2 - 2 \sqrt{6}$

$x = - 2 \pm 2 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 2 \pm 2 \sqrt{6}$

Decimal Form:

$x = 2.89897948 \dots, - 6.89897948 \dots$

Step 4