Algebra Examples

$\frac{1}{2} x^{2} - \frac{x}{4} - 1 = 0$

Step 1

Simplify the left side.

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Step 1.1

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

$\frac{x^{2}}{2} - \frac{x}{4} - 1 = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

$4 (\frac{x^{2}}{2}) + 4 (- \frac{x}{4}) + 4 \cdot - 1 = 0$

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

Factor $2$ out of $4$ .

$2 (2) (\frac{x^{2}}{2}) + 4 (- \frac{x}{4}) + 4 \cdot - 1 = 0$

Step 2.2.1.2

Cancel the common factor.

$2 \cdot (2 (\frac{x^{2}}{2})) + 4 (- \frac{x}{4}) + 4 \cdot - 1 = 0$

Step 2.2.1.3

Rewrite the expression.

$2 x^{2} + 4 (- \frac{x}{4}) + 4 \cdot - 1 = 0$

Step 2.2.2

Cancel the common factor of $4$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$2 x^{2} + 4 (\frac{- x}{4}) + 4 \cdot - 1 = 0$

Step 2.2.2.2

Cancel the common factor.

$2 x^{2} + 4 (\frac{- x}{4}) + 4 \cdot - 1 = 0$

Step 2.2.2.3

Rewrite the expression.

$2 x^{2} - x + 4 \cdot - 1 = 0$

Step 2.2.3

Multiply $4$ by $- 1$ .

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

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 4$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 4}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 32}}{2 \cdot 2}$

Step 5.1.3

Add $1$ and $32$ .

$x = \frac{1 \pm \sqrt{33}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{33}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{33}}{4}$

Decimal Form:

$x = 1.68614066 \dots, - 1.18614066 \dots$