Find the Quadratic Equation 3 , 5/6

$3$ , $\frac{5}{6}$

Step 1

$x = 3$ and $x = \frac{5}{6}$ are the two real distinct solutions for the quadratic equation, which means that $x - 3$ and $x - \frac{5}{6}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - 3) (x - \frac{5}{6})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

$x^{2} - \frac{5 x}{6} - 3 x + 3 (- 1) (\frac{- 5}{6}) = 0$

Factor $3$ out of $6$ .

$x^{2} - \frac{5 x}{6} - 3 x + 3 \cdot (- 1 \frac{- 5}{3 \cdot 2}) = 0$

Cancel the common factor.

$x^{2} - \frac{5 x}{6} - 3 x + 3 \cdot (- 1 \frac{- 5}{3 \cdot 2}) = 0$

Rewrite the expression.

$x^{2} - \frac{5 x}{6} - 3 x - 1 (\frac{- 5}{2}) = 0$

Move the negative in front of the fraction.

$x^{2} - \frac{5 x}{6} - 3 x - 1 (- \frac{5}{2}) = 0$

Multiply $- 1 (- \frac{5}{2})$ .

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{5 x}{6} - 3 x + 1 (\frac{5}{2}) = 0$

Multiply $\frac{5}{2}$ by $1$ .

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

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Combine $- 3 x$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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

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

Multiply $6$ by $- 3$ .

$\frac{6 x^{2} - 5 x - 18 x + 5 \cdot 3}{6} = 0$

Multiply $5$ by $3$ .

$\frac{6 x^{2} - 5 x - 18 x + 15}{6} = 0$

Subtract $18 x$ from $- 5 x$ .

$\frac{6 x^{2} - 23 x + 15}{6} = 0$

Factor by grouping.

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 $a \cdot c = 6 \cdot 15 = 90$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$\frac{6 x^{2} - 23 x + 15}{6} = 0$

Rewrite $- 23$ as $- 5$ plus $- 18$

$\frac{6 x^{2} + (- 5 - 18) x + 15}{6} = 0$

Apply the distributive property.

$\frac{6 x^{2} - 5 x - 18 x + 15}{6} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(6 x^{2} - 5 x) - 18 x + 15}{6} = 0$

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

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

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

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

Step 5

Expand $(6 x - 5) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 3$ by $6$ .

$\frac{6 x^{2} - 18 x - 5 x - 5 \cdot - 3}{6} = 0$

Multiply $- 5$ by $- 3$ .

$\frac{6 x^{2} - 18 x - 5 x + 15}{6} = 0$

Subtract $5 x$ from $- 18 x$ .

$\frac{6 x^{2} - 23 x + 15}{6} = 0$

Step 7

Split the fraction $\frac{6 x^{2} - 23 x + 15}{6}$ into two fractions.

$\frac{6 x^{2} - 23 x}{6} + \frac{15}{6} = 0$

Step 8

Split the fraction $\frac{6 x^{2} - 23 x}{6}$ into two fractions.

$\frac{6 x^{2}}{6} + \frac{- 23 x}{6} + \frac{15}{6} = 0$

Step 9

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{2}}{6} + \frac{- 23 x}{6} + \frac{15}{6} = 0$

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

$x^{2} + \frac{- 23 x}{6} + \frac{15}{6} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{23 x}{6} + \frac{15}{6} = 0$

Step 11

Cancel the common factor of $15$ and $6$ .

Factor $3$ out of $15$ .

$x^{2} - \frac{23 x}{6} + \frac{3 (5)}{6} = 0$

Cancel the common factors.

Factor $3$ out of $6$ .

$x^{2} - \frac{23 x}{6} + \frac{3 \cdot 5}{3 \cdot 2} = 0$

Cancel the common factor.

$x^{2} - \frac{23 x}{6} + \frac{3 \cdot 5}{3 \cdot 2} = 0$

Rewrite the expression.

$x^{2} - \frac{23 x}{6} + \frac{5}{2} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${3, \frac{5}{6}}$ is $y = x^{2} - \frac{23 x}{6} + \frac{5}{2}$ .

$y = x^{2} - \frac{23 x}{6} + \frac{5}{2}$

Step 13

Find the Quadratic Equation 3 , 5/6

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