Algebra Examples

$- \frac{1}{4}$ , $- \frac{1}{6}$

Step 1

$x = - \frac{1}{4}$ and $x = - \frac{1}{6}$ are the two real distinct solutions for the quadratic equation, which means that $x + \frac{1}{4}$ and $x + \frac{1}{6}$ are the factors of the quadratic equation.

$(x + \frac{1}{4}) (x + \frac{1}{6}) = 0$

Step 2

Expand $(x + \frac{1}{4}) (x + \frac{1}{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x + \frac{1}{6}) + \frac{1}{4} \cdot (x + \frac{1}{6}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (\frac{1}{6}) + \frac{1}{4} \cdot (x + \frac{1}{6}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (\frac{1}{6}) + \frac{1}{4} x + \frac{1}{4} \cdot \frac{1}{6} = 0$

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 $x$ .

$x^{2} + x (\frac{1}{6}) + \frac{1}{4} x + \frac{1}{4} \cdot \frac{1}{6} = 0$

Step 3.1.2

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

$x^{2} + \frac{x}{6} + \frac{1}{4} x + \frac{1}{4} \cdot \frac{1}{6} = 0$

Step 3.1.3

Combine $\frac{1}{4}$ and $x$ .

$x^{2} + \frac{x}{6} + \frac{x}{4} + \frac{1}{4} \cdot \frac{1}{6} = 0$

Step 3.1.4

Multiply $\frac{1}{4} \cdot \frac{1}{6}$ .

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Step 3.1.4.1

Multiply $\frac{1}{4}$ by $\frac{1}{6}$ .

$x^{2} + \frac{x}{6} + \frac{x}{4} + \frac{1}{4 \cdot 6} = 0$

Step 3.1.4.2

Multiply $4$ by $6$ .

$x^{2} + \frac{x}{6} + \frac{x}{4} + \frac{1}{24} = 0$

Step 3.2

To write $\frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + \frac{x}{6} \cdot \frac{2}{2} + \frac{x}{4} + \frac{1}{24} = 0$

Step 3.3

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} + \frac{x}{6} \cdot \frac{2}{2} + \frac{x}{4} \cdot \frac{3}{3} + \frac{1}{24} = 0$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{6}$ by $\frac{2}{2}$ .

$x^{2} + \frac{x \cdot 2}{6 \cdot 2} + \frac{x}{4} \cdot \frac{3}{3} + \frac{1}{24} = 0$

Step 3.4.2

Multiply $6$ by $2$ .

$x^{2} + \frac{x \cdot 2}{12} + \frac{x}{4} \cdot \frac{3}{3} + \frac{1}{24} = 0$

Step 3.4.3

Multiply $\frac{x}{4}$ by $\frac{3}{3}$ .

$x^{2} + \frac{x \cdot 2}{12} + \frac{x \cdot 3}{4 \cdot 3} + \frac{1}{24} = 0$

Step 3.4.4

Multiply $4$ by $3$ .

$x^{2} + \frac{x \cdot 2}{12} + \frac{x \cdot 3}{12} + \frac{1}{24} = 0$

Step 3.5

Combine the numerators over the common denominator.

$x^{2} + \frac{x \cdot 2 + x \cdot 3}{12} + \frac{1}{24} = 0$

Step 4

Reorder $x$ and $3$ .

$x^{2} + \frac{2 \cdot x + 3 \cdot x}{12} + \frac{1}{24} = 0$

Step 5

Add $2 \cdot x$ and $3 \cdot x$ .

$x^{2} + \frac{5 x}{12} + \frac{1}{24} = 0$

Step 6

The standard quadratic equation using the given set of solutions ${- \frac{1}{4}, - \frac{1}{6}}$ is $y = x^{2} + \frac{5 x}{12} + \frac{1}{24}$ .

$y = x^{2} + \frac{5 x}{12} + \frac{1}{24}$

Step 7