Algebra Examples

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

Step 1

$x = \frac{1}{2}$ and $x = \frac{1}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - \frac{1}{2}$ and $x - \frac{1}{3}$ are the factors of the quadratic equation.

$(x - \frac{1}{2}) (x - \frac{1}{3}) = 0$

Step 2

Expand $(x - \frac{1}{2}) (x - \frac{1}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - \frac{1}{3}) - \frac{1}{2} \cdot (x - \frac{1}{3}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (- \frac{1}{3}) - \frac{1}{2} \cdot (x - \frac{1}{3}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (- \frac{1}{3}) - \frac{1}{2} x - \frac{1}{2} \cdot (- \frac{1}{3}) = 0$

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 $x$ .

$x^{2} + x (- \frac{1}{3}) - \frac{1}{2} x - \frac{1}{2} \cdot (- \frac{1}{3}) = 0$

Step 3.1.2

Combine $x$ and $\frac{1}{3}$ .

$x^{2} - \frac{x}{3} - \frac{1}{2} x - \frac{1}{2} \cdot (- \frac{1}{3}) = 0$

Step 3.1.3

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

$x^{2} - \frac{x}{3} - \frac{x}{2} - \frac{1}{2} \cdot (- \frac{1}{3}) = 0$

Step 3.1.4

Multiply $- \frac{1}{2} (- \frac{1}{3})$ .

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{x}{3} - \frac{x}{2} + 1 (\frac{1}{2}) \cdot \frac{1}{3} = 0$

Step 3.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 3.1.4.3

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 3.1.4.4

Multiply $2$ by $3$ .

$x^{2} - \frac{x}{3} - \frac{x}{2} + \frac{1}{6} = 0$

Step 3.2

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

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

Step 3.4.2

Multiply $3$ by $2$ .

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

Step 3.4.3

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

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

Step 3.4.4

Multiply $2$ by $3$ .

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

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

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

Step 3.7

Combine $x^{2}$ and $\frac{6}{6}$ .

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

Step 3.8

Combine the numerators over the common denominator.

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

Step 3.9

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Reorder terms.

$\frac{6 x^{2} - 1 \cdot (2 x) - 1 \cdot (3 x) + 1}{6} = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(6 x^{2} - 1 \cdot (2 x)) - 1 \cdot (3 x) + 1}{6} = 0$

Step 4.2.2

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

$\frac{2 x (3 x - 1) - (3 x - 1)}{6} = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$\frac{(3 x - 1) (2 x - 1)}{6} = 0$

Step 5

Expand $(3 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{3 x (2 x - 1) - 1 (2 x - 1)}{6} = 0$

Step 5.2

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 1 - 1 (2 x - 1)}{6} = 0$

Step 5.3

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot (2 x \cdot x) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6.1.2

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

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Step 6.1.2.1

Move $x$ .

$\frac{3 \cdot (2 (x \cdot x)) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6.1.2.2

Multiply $x$ by $x$ .

$\frac{3 \cdot (2 x^{2}) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6.1.3

Multiply $3$ by $2$ .

$\frac{6 x^{2} + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6.1.4

Multiply $- 1$ by $3$ .

$\frac{6 x^{2} - 3 x - 1 (2 x) - 1 \cdot - 1}{6} = 0$

Step 6.1.5

Multiply $2$ by $- 1$ .

$\frac{6 x^{2} - 3 x - 2 x - 1 \cdot - 1}{6} = 0$

Step 6.1.6

Multiply $- 1$ by $- 1$ .

$\frac{6 x^{2} - 3 x - 2 x + 1}{6} = 0$

Step 6.2

Subtract $2 x$ from $- 3 x$ .

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

Step 7

Split the fraction $\frac{6 x^{2} - 5 x + 1}{6}$ into two fractions.

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

Step 8

Split the fraction $\frac{6 x^{2} - 5 x}{6}$ into two fractions.

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

Step 9

Cancel the common factor of $6$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

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

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

Step 10

Move the negative in front of the fraction.

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

Step 11

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

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

Step 12