Algebra Examples

$x^{2} = \frac{5}{6} x - \frac{1}{6}$

Step 1

Combine $\frac{5}{6}$ and $x$ .

$x^{2} = \frac{5 x}{6} - \frac{1}{6}$

Step 2

Subtract $\frac{5 x}{6}$ from both sides of the equation.

$x^{2} - \frac{5 x}{6} = - \frac{1}{6}$

Step 3

Add $\frac{1}{6}$ to both sides of the equation.

$x^{2} - \frac{5 x}{6} + \frac{1}{6} = 0$

Step 4

Multiply through by the least common denominator $6$ , then simplify.

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Step 4.1

Apply the distributive property.

$6 x^{2} + 6 (- \frac{5 x}{6}) + 6 (\frac{1}{6}) = 0$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of $6$ .

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Step 4.2.1.1

Move the leading negative in $- \frac{5 x}{6}$ into the numerator.

$6 x^{2} + 6 (\frac{- 5 x}{6}) + 6 (\frac{1}{6}) = 0$

Step 4.2.1.2

Cancel the common factor.

$6 x^{2} + 6 (\frac{- 5 x}{6}) + 6 (\frac{1}{6}) = 0$

Step 4.2.1.3

Rewrite the expression.

$6 x^{2} - 5 x + 6 (\frac{1}{6}) = 0$

Step 4.2.2

Cancel the common factor of $6$ .

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Step 4.2.2.1

Cancel the common factor.

$6 x^{2} - 5 x + 6 (\frac{1}{6}) = 0$

Step 4.2.2.2

Rewrite the expression.

$6 x^{2} - 5 x + 1 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 6$ , $b = - 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (6 \cdot 1)}}{2 \cdot 6}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot 1}}{2 \cdot 6}$

Step 7.1.2

Multiply $- 4 \cdot 6 \cdot 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{5 \pm \sqrt{25 - 24 \cdot 1}}{2 \cdot 6}$

Step 7.1.2.2

Multiply $- 24$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 24}}{2 \cdot 6}$

Step 7.1.3

Subtract $24$ from $25$ .

$x = \frac{5 \pm \sqrt{1}}{2 \cdot 6}$

Step 7.1.4

Any root of $1$ is $1$ .

$x = \frac{5 \pm 1}{2 \cdot 6}$

Step 7.2

Multiply $2$ by $6$ .

$x = \frac{5 \pm 1}{12}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1}{2}, \frac{1}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, \frac{1}{3}$

Decimal Form:

$x = 0.5, 0.\overline{3}$