Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Multiply $- 1$ by $2$ .

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

Step 2.2.3

Multiply $2$ by $- 2$ .

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

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 $a = 1$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.1.2.2

Multiply $- 4$ by $- 4$ .

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

Step 5.1.3

Add $4$ and $16$ .

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

Step 5.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 5.1.4.1

Factor $4$ out of $20$ .

$x = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{5}}{2}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{2}$ .

$x = 1 \pm \sqrt{5}$

Step 6

The final answer is the combination of both solutions.

$x = 1 + \sqrt{5}, 1 - \sqrt{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{5}, 1 - \sqrt{5}$

Decimal Form:

$x = 3.23606797 \dots, - 1.23606797 \dots$