Finite Math Examples

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

Step 1

Set $- \frac{1}{2} x^{2} - \frac{1}{2} x + \frac{3}{2}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

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.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

Cancel the common factor.

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

Step 2.2.2.1.3

Rewrite the expression.

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

Step 2.2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.3

Cancel the common factor of $2$ .

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Step 2.2.2.3.1

Cancel the common factor.

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

Step 2.2.2.3.2

Rewrite the expression.

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

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = - 1$ , $b = - 1$ , and $c = 3$ into the quadratic formula and solve for $x$ .

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

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 $- 1$ to the power of $2$ .

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

Step 2.5.1.2

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

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Step 2.5.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 2.5.1.2.2

Multiply $4$ by $3$ .

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

Step 2.5.1.3

Add $1$ and $12$ .

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

Step 2.5.2

Multiply $2$ by $- 1$ .

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

Step 2.5.3

Move the negative in front of the fraction.

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

Step 2.6

The final answer is the combination of both solutions.

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

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Step 4