Finite Math Examples

y = - \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2}

$y = - \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2}$

Step 1

Set

- \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2}

$- \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2}$ equal to

0

$0$ .

- \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2} = 0

$- \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2} = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine

x^{2}

$x^{2}$ and

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{x^{2}}{2} - \frac{1}{2} x + \frac{3}{2} = 0

$- \frac{x^{2}}{2} - \frac{1}{2} x + \frac{3}{2} = 0$

Step 2.1.2

Combine

x

$x$ and

\frac{1}{2}

$\frac{1}{2}$ .

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

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

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

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

Step 2.2

Multiply through by the least common denominator

2

$2$ , then simplify.

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Step 2.2.1

Apply the distributive property.

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

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.2.1.1

Move the leading negative in

- \frac{x^{2}}{2}

$- \frac{x^{2}}{2}$ into the numerator.

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

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

Step 2.2.2.1.2

Cancel the common factor.

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

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

Step 2.2.2.1.3

Rewrite the expression.

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

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

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

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

Step 2.2.2.2

Cancel the common factor of

2

$2$ .

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Step 2.2.2.2.1

Move the leading negative in

- \frac{x}{2}

$- \frac{x}{2}$ into the numerator.

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

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

Step 2.2.2.2.3

Rewrite the expression.

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

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

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

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

Step 2.2.2.3

Cancel the common factor of

2

$2$ .

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Step 2.2.2.3.1

Cancel the common factor.

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

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

Step 2.2.2.3.2

Rewrite the expression.

- x^{2} - x + 3 = 0

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

- x^{2} - x + 3 = 0

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

- x^{2} - x + 3 = 0

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

- x^{2} - x + 3 = 0

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

Step 2.3

Use the quadratic formula to find the solutions.

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

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

Step 2.4

Substitute the values

a = - 1

$a = - 1$ ,

b = - 1

$b = - 1$ , and

c = 3

$c = 3$ into the quadratic formula and solve for

x

$x$ .

\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 1 \cdot 3)}}{2 \cdot - 1}

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 1 \cdot 3)}}{2 \cdot - 1}$

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

- 1

$- 1$ to the power of

2

$2$ .

x = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 3}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 3}}{2 \cdot - 1}$

Step 2.5.1.2

Multiply

- 4 \cdot - 1 \cdot 3

$- 4 \cdot - 1 \cdot 3$ .

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Step 2.5.1.2.1

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

x = \frac{1 \pm \sqrt{1 + 4 \cdot 3}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{1 + 4 \cdot 3}}{2 \cdot - 1}$

Step 2.5.1.2.2

Multiply

4

$4$ by

3

$3$ .

x = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot - 1}$

x = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot - 1}$

Step 2.5.1.3

Add

1

$1$ and

12

$12$ .

x = \frac{1 \pm \sqrt{13}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{13}}{2 \cdot - 1}$

x = \frac{1 \pm \sqrt{13}}{2 \cdot - 1}

$x = \frac{1 \pm \sqrt{13}}{2 \cdot - 1}$

Step 2.5.2

Multiply

2

$2$ by

- 1

$- 1$ .

x = \frac{1 \pm \sqrt{13}}{- 2}

$x = \frac{1 \pm \sqrt{13}}{- 2}$

Step 2.5.3

Move the negative in front of the fraction.

x = - \frac{1 \pm \sqrt{13}}{2}

$x = - \frac{1 \pm \sqrt{13}}{2}$

x = - \frac{1 \pm \sqrt{13}}{2}

$x = - \frac{1 \pm \sqrt{13}}{2}$

Step 2.6

The final answer is the combination of both solutions.

x = - \frac{1 + \sqrt{13}}{2}, - \frac{1 - \sqrt{13}}{2}

$x = - \frac{1 + \sqrt{13}}{2}, - \frac{1 - \sqrt{13}}{2}$

x = - \frac{1 \pm \sqrt{13}}{2}

$x = - \frac{1 \pm \sqrt{13}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

x = - \frac{1 \pm \sqrt{13}}{2}

$x = - \frac{1 \pm \sqrt{13}}{2}$

Decimal Form:

x = - 2.30277563 \dots, 1.30277563 \dots

$x = - 2.30277563 \dots, 1.30277563 \dots$

Step 4