Algebra Examples

${4, \frac{5}{6}}$

Step 1

$x = 4$ and $x = \frac{5}{6}$ are the two real distinct solutions for the quadratic equation, which means that $x - 4$ and $x - \frac{5}{6}$ are the factors of the quadratic equation.

$(x - 4) (x - \frac{5}{6}) = 0$

Step 2

Expand $(x - 4) (x - \frac{5}{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - \frac{5}{6}) - 4 (x - \frac{5}{6}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (- \frac{5}{6}) - 4 (x - \frac{5}{6}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (- \frac{5}{6}) - 4 x - 4 (- \frac{5}{6}) = 0$

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 $x$ .

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

Step 3.1.2

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

$x^{2} - \frac{x \cdot 5}{6} - 4 x - 4 (- \frac{5}{6}) = 0$

Step 3.1.3

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

$x^{2} - \frac{5 \cdot x}{6} - 4 x - 4 (- \frac{5}{6}) = 0$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

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

$x^{2} - \frac{5 x}{6} - 4 x - 4 (\frac{- 5}{6}) = 0$

Step 3.1.4.2

Factor $2$ out of $- 4$ .

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

Step 3.1.4.3

Factor $2$ out of $6$ .

$x^{2} - \frac{5 x}{6} - 4 x + 2 \cdot (- 2 \frac{- 5}{2 \cdot 3}) = 0$

Step 3.1.4.4

Cancel the common factor.

$x^{2} - \frac{5 x}{6} - 4 x + 2 \cdot (- 2 \frac{- 5}{2 \cdot 3}) = 0$

Step 3.1.4.5

Rewrite the expression.

$x^{2} - \frac{5 x}{6} - 4 x - 2 (\frac{- 5}{3}) = 0$

Step 3.1.5

Combine $- 2$ and $\frac{- 5}{3}$ .

$x^{2} - \frac{5 x}{6} - 4 x + \frac{- 2 \cdot - 5}{3} = 0$

Step 3.1.6

Multiply $- 2$ by $- 5$ .

$x^{2} - \frac{5 x}{6} - 4 x + \frac{10}{3} = 0$

Step 3.2

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x^{2} - \frac{5 x}{6} - 4 x \cdot \frac{6}{6} + \frac{10}{3} = 0$

Step 3.3

Combine $- 4 x$ and $\frac{6}{6}$ .

$x^{2} - \frac{5 x}{6} + \frac{- 4 x \cdot 6}{6} + \frac{10}{3} = 0$

Step 3.4

Combine the numerators over the common denominator.

$x^{2} + \frac{- 5 x - 4 x \cdot 6}{6} + \frac{10}{3} = 0$

Step 3.5

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

$x^{2} \cdot \frac{6}{6} + \frac{- 5 x - 4 x \cdot 6}{6} + \frac{10}{3} = 0$

Step 3.6

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

$\frac{x^{2} \cdot 6}{6} + \frac{- 5 x - 4 x \cdot 6}{6} + \frac{10}{3} = 0$

Step 3.7

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 6 - 5 x - 4 x \cdot 6}{6} + \frac{10}{3} = 0$

Step 3.8

To write $\frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 6 - 5 x - 4 x \cdot 6}{6} + \frac{10}{3} \cdot \frac{2}{2} = 0$

Step 3.9

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

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Step 3.9.1

Multiply $\frac{10}{3}$ by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 6 - 5 x - 4 x \cdot 6}{6} + \frac{10 \cdot 2}{3 \cdot 2} = 0$

Step 3.9.2

Multiply $3$ by $2$ .

$\frac{x^{2} \cdot 6 - 5 x - 4 x \cdot 6}{6} + \frac{10 \cdot 2}{6} = 0$

Step 3.10

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 6 - 5 x - 4 x \cdot 6 + 10 \cdot 2}{6} = 0$

Step 4

Simplify the numerator.

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Step 4.1

Move $6$ to the left of $x^{2}$ .

$\frac{6 \cdot x^{2} - 5 x - 4 x \cdot 6 + 10 \cdot 2}{6} = 0$

Step 4.2

Multiply $6$ by $- 4$ .

$\frac{6 x^{2} - 5 x - 24 x + 10 \cdot 2}{6} = 0$

Step 4.3

Multiply $10$ by $2$ .

$\frac{6 x^{2} - 5 x - 24 x + 20}{6} = 0$

Step 4.4

Subtract $24 x$ from $- 5 x$ .

$\frac{6 x^{2} - 29 x + 20}{6} = 0$

Step 4.5

Factor by grouping.

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Step 4.5.1

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 $a \cdot c = 6 \cdot 20 = 120$ and whose sum is $b = - 29$ .

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Step 4.5.1.1

Factor $- 29$ out of $- 29 x$ .

$\frac{6 x^{2} - 29 x + 20}{6} = 0$

Step 4.5.1.2

Rewrite $- 29$ as $- 5$ plus $- 24$

$\frac{6 x^{2} + (- 5 - 24) x + 20}{6} = 0$

Step 4.5.1.3

Apply the distributive property.

$\frac{6 x^{2} - 5 x - 24 x + 20}{6} = 0$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(6 x^{2} - 5 x) - 24 x + 20}{6} = 0$

Step 4.5.2.2

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

$\frac{x (6 x - 5) - 4 (6 x - 5)}{6} = 0$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $6 x - 5$ .

$\frac{(6 x - 5) (x - 4)}{6} = 0$

Step 5

Expand $(6 x - 5) (x - 4)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{6 x (x - 4) - 5 (x - 4)}{6} = 0$

Step 5.2

Apply the distributive property.

$\frac{6 x \cdot x + 6 x \cdot - 4 - 5 (x - 4)}{6} = 0$

Step 5.3

Apply the distributive property.

$\frac{6 x \cdot x + 6 x \cdot - 4 - 5 x - 5 \cdot - 4}{6} = 0$

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

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

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Step 6.1.1.1

Move $x$ .

$\frac{6 (x \cdot x) + 6 x \cdot - 4 - 5 x - 5 \cdot - 4}{6} = 0$

Step 6.1.1.2

Multiply $x$ by $x$ .

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

Step 6.1.2

Multiply $- 4$ by $6$ .

$\frac{6 x^{2} - 24 x - 5 x - 5 \cdot - 4}{6} = 0$

Step 6.1.3

Multiply $- 5$ by $- 4$ .

$\frac{6 x^{2} - 24 x - 5 x + 20}{6} = 0$

Step 6.2

Subtract $5 x$ from $- 24 x$ .

$\frac{6 x^{2} - 29 x + 20}{6} = 0$

Step 7

Split the fraction $\frac{6 x^{2} - 29 x + 20}{6}$ into two fractions.

$\frac{6 x^{2} - 29 x}{6} + \frac{20}{6} = 0$

Step 8

Split the fraction $\frac{6 x^{2} - 29 x}{6}$ into two fractions.

$\frac{6 x^{2}}{6} + \frac{- 29 x}{6} + \frac{20}{6} = 0$

Step 9

Cancel the common factor of $6$ .

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Step 9.1

Cancel the common factor.

$\frac{6 x^{2}}{6} + \frac{- 29 x}{6} + \frac{20}{6} = 0$

Step 9.2

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

$x^{2} + \frac{- 29 x}{6} + \frac{20}{6} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{29 x}{6} + \frac{20}{6} = 0$

Step 11

Cancel the common factor of $20$ and $6$ .

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Step 11.1

Factor $2$ out of $20$ .

$x^{2} - \frac{29 x}{6} + \frac{2 (10)}{6} = 0$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $6$ .

$x^{2} - \frac{29 x}{6} + \frac{2 \cdot 10}{2 \cdot 3} = 0$

Step 11.2.2

Cancel the common factor.

$x^{2} - \frac{29 x}{6} + \frac{2 \cdot 10}{2 \cdot 3} = 0$

Step 11.2.3

Rewrite the expression.

$x^{2} - \frac{29 x}{6} + \frac{10}{3} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${4, \frac{5}{6}}$ is $y = x^{2} - \frac{29 x}{6} + \frac{10}{3}$ .

$y = x^{2} - \frac{29 x}{6} + \frac{10}{3}$

Step 13