Algebra Examples

$- \frac{1}{6}$ , $\frac{5}{6}$

Step 1

$x = - \frac{1}{6}$ and $x = \frac{5}{6}$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{1}{6}$ and $x - \frac{5}{6}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x + \frac{1}{6}) (x - \frac{5}{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Move $5$ to the left of $x$ .

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

Step 3.1.4

Combine $\frac{1}{6}$ and $x$ .

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

Step 3.1.5

Multiply $\frac{1}{6} (- \frac{5}{6})$ .

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Step 3.1.5.1

Multiply $\frac{1}{6}$ by $\frac{5}{6}$ .

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

Step 3.1.5.2

Multiply $6$ by $6$ .

$x^{2} - \frac{5 x}{6} + \frac{x}{6} - \frac{5}{36} = 0$

Step 3.2

Combine the numerators over the common denominator.

$x^{2} + \frac{- 5 x + x}{6} - \frac{5}{36} = 0$

Step 4

Add $- 5 x$ and $x$ .

$x^{2} + \frac{- 4 x}{6} - \frac{5}{36} = 0$

Step 5

Simplify each term.

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Step 5.1

Cancel the common factor of $- 4$ and $6$ .

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Step 5.1.1

Factor $2$ out of $- 4 x$ .

$x^{2} + \frac{2 (- 2 x)}{6} - \frac{5}{36} = 0$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $2$ out of $6$ .

$x^{2} + \frac{2 (- 2 x)}{2 (3)} - \frac{5}{36} = 0$

Step 5.1.2.2

Cancel the common factor.

$x^{2} + \frac{2 (- 2 x)}{2 \cdot 3} - \frac{5}{36} = 0$

Step 5.1.2.3

Rewrite the expression.

$x^{2} + \frac{- 2 x}{3} - \frac{5}{36} = 0$

Step 5.2

Move the negative in front of the fraction.

$x^{2} - \frac{2 x}{3} - \frac{5}{36} = 0$

Step 6

The standard quadratic equation using the given set of solutions ${- \frac{1}{6}, \frac{5}{6}}$ is $y = x^{2} - \frac{2 x}{3} - \frac{5}{36}$ .

$y = x^{2} - \frac{2 x}{3} - \frac{5}{36}$

Step 7