Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

Combine $x$ and $\frac{1}{5}$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Multiply $6$ by $5$ .

$x^{2} + \frac{x}{5} + \frac{x}{6} + \frac{1}{30} = 0$

Step 3.2

To write $\frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x^{2} + \frac{x}{5} \cdot \frac{6}{6} + \frac{x}{6} + \frac{1}{30} = 0$

Step 3.3

To write $\frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{2} + \frac{x}{5} \cdot \frac{6}{6} + \frac{x}{6} \cdot \frac{5}{5} + \frac{1}{30} = 0$

Step 3.4

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

$x^{2} + \frac{x \cdot 6}{5 \cdot 6} + \frac{x}{6} \cdot \frac{5}{5} + \frac{1}{30} = 0$

Step 3.4.2

Multiply $5$ by $6$ .

$x^{2} + \frac{x \cdot 6}{30} + \frac{x}{6} \cdot \frac{5}{5} + \frac{1}{30} = 0$

Step 3.4.3

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

$x^{2} + \frac{x \cdot 6}{30} + \frac{x \cdot 5}{6 \cdot 5} + \frac{1}{30} = 0$

Step 3.4.4

Multiply $6$ by $5$ .

$x^{2} + \frac{x \cdot 6}{30} + \frac{x \cdot 5}{30} + \frac{1}{30} = 0$

Step 3.5

Combine the numerators over the common denominator.

$x^{2} + \frac{x \cdot 6 + x \cdot 5}{30} + \frac{1}{30} = 0$

Step 4

Combine the numerators over the common denominator.

$x^{2} + \frac{x \cdot 6 + x \cdot 5 + 1}{30} = 0$

Step 5

Simplify each term.

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Step 5.1

Move $6$ to the left of $x$ .

$x^{2} + \frac{6 \cdot x + x \cdot 5 + 1}{30} = 0$

Step 5.2

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

$x^{2} + \frac{6 x + 5 x + 1}{30} = 0$

Step 6

Add $6 x$ and $5 x$ .

$x^{2} + \frac{11 x + 1}{30} = 0$

Step 7

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$x^{2} \cdot \frac{30}{30} + \frac{11 x + 1}{30} = 0$

Step 8

Combine $x^{2}$ and $\frac{30}{30}$ .

$\frac{x^{2} \cdot 30}{30} + \frac{11 x + 1}{30} = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 30 + 11 x + 1}{30} = 0$

Step 10

Factor by grouping.

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Step 10.1

Reorder terms.

$\frac{30 x^{2} + 11 x + 1}{30} = 0$

Step 10.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 30 \cdot 1 = 30$ and whose sum is $b = 11$ .

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Step 10.2.1

Factor $11$ out of $11 x$ .

$\frac{30 x^{2} + 11 (x) + 1}{30} = 0$

Step 10.2.2

Rewrite $11$ as $5$ plus $6$

$\frac{30 x^{2} + (5 + 6) x + 1}{30} = 0$

Step 10.2.3

Apply the distributive property.

$\frac{30 x^{2} + 5 x + 6 x + 1}{30} = 0$

Step 10.3

Factor out the greatest common factor from each group.

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Step 10.3.1

Group the first two terms and the last two terms.

$\frac{(30 x^{2} + 5 x) + 6 x + 1}{30} = 0$

Step 10.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{5 x (6 x + 1) + 1 (6 x + 1)}{30} = 0$

Step 10.4

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$\frac{(6 x + 1) (5 x + 1)}{30} = 0$

Step 11

Expand $(6 x + 1) (5 x + 1)$ using the FOIL Method.

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Step 11.1

Apply the distributive property.

$\frac{6 x (5 x + 1) + 1 (5 x + 1)}{30} = 0$

Step 11.2

Apply the distributive property.

$\frac{6 x (5 x) + 6 x \cdot 1 + 1 (5 x + 1)}{30} = 0$

Step 11.3

Apply the distributive property.

$\frac{6 x (5 x) + 6 x \cdot 1 + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12

Simplify and combine like terms.

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Step 12.1

Simplify each term.

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Step 12.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 \cdot (5 x \cdot x) + 6 x \cdot 1 + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 12.1.2.1

Move $x$ .

$\frac{6 \cdot (5 (x \cdot x)) + 6 x \cdot 1 + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12.1.2.2

Multiply $x$ by $x$ .

$\frac{6 \cdot (5 x^{2}) + 6 x \cdot 1 + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12.1.3

Multiply $6$ by $5$ .

$\frac{30 x^{2} + 6 x \cdot 1 + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12.1.4

Multiply $6$ by $1$ .

$\frac{30 x^{2} + 6 x + 1 (5 x) + 1 \cdot 1}{30} = 0$

Step 12.1.5

Multiply $5 x$ by $1$ .

$\frac{30 x^{2} + 6 x + 5 x + 1 \cdot 1}{30} = 0$

Step 12.1.6

Multiply $1$ by $1$ .

$\frac{30 x^{2} + 6 x + 5 x + 1}{30} = 0$

Step 12.2

Add $6 x$ and $5 x$ .

$\frac{30 x^{2} + 11 x + 1}{30} = 0$

Step 13

Split the fraction $\frac{30 x^{2} + 11 x + 1}{30}$ into two fractions.

$\frac{30 x^{2} + 11 x}{30} + \frac{1}{30} = 0$

Step 14

Split the fraction $\frac{30 x^{2} + 11 x}{30}$ into two fractions.

$\frac{30 x^{2}}{30} + \frac{11 x}{30} + \frac{1}{30} = 0$

Step 15

Cancel the common factor of $30$ .

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Step 15.1

Cancel the common factor.

$\frac{30 x^{2}}{30} + \frac{11 x}{30} + \frac{1}{30} = 0$

Step 15.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{11 x}{30} + \frac{1}{30} = 0$

Step 16

The standard quadratic equation using the given set of solutions ${- \frac{1}{6}, - \frac{1}{5}}$ is $y = x^{2} + \frac{11 x}{30} + \frac{1}{30}$ .

$y = x^{2} + \frac{11 x}{30} + \frac{1}{30}$

Step 17