Find the Quadratic Equation Given the Roots 2/3 , 7

$\frac{2}{3}$ , $7$

$x = \frac{2}{3}$ and $x = 7$ are the two real distinct solutions for the quadratic equation, which means that $x - \frac{2}{3}$ and $x - 7$ are the factors of the quadratic equation.

$(x - \frac{2}{3}) (x - 7) = 0$

Expand $(x - \frac{2}{3}) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$x (x - 7) - \frac{2}{3} \cdot (x - 7) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - \frac{2}{3} \cdot (x - 7) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - \frac{2}{3} x - \frac{2}{3} \cdot - 7 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 7 - \frac{2}{3} x - \frac{2}{3} \cdot - 7 = 0$

Move $- 7$ to the left of $x$ .

$x^{2} - 7 \cdot x - \frac{2}{3} x - \frac{2}{3} \cdot - 7 = 0$

Combine $x$ and $\frac{2}{3}$ .

$x^{2} - 7 x - \frac{x \cdot 2}{3} - \frac{2}{3} \cdot - 7 = 0$

Move $2$ to the left of $x$ .

$x^{2} - 7 x - \frac{2 \cdot x}{3} - \frac{2}{3} \cdot - 7 = 0$

Multiply $- \frac{2}{3} \cdot - 7$ .

Multiply $- 7$ by $- 1$ .

$x^{2} - 7 x - \frac{2 x}{3} + 7 (\frac{2}{3}) = 0$

Combine $7$ and $\frac{2}{3}$ .

$x^{2} - 7 x - \frac{2 x}{3} + \frac{7 \cdot 2}{3} = 0$

Multiply $7$ by $2$ .

$x^{2} - 7 x - \frac{2 x}{3} + \frac{14}{3} = 0$

To write $- 7 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - 7 x \cdot \frac{3}{3} - \frac{2 x}{3} + \frac{14}{3} = 0$

Combine $- 7 x$ and $\frac{3}{3}$ .

$x^{2} + \frac{- 7 x \cdot 3}{3} - \frac{2 x}{3} + \frac{14}{3} = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{- 7 x \cdot 3 - 2 x}{3} + \frac{14}{3} = 0$

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

$x^{2} \cdot \frac{3}{3} + \frac{- 7 x \cdot 3 - 2 x}{3} + \frac{14}{3} = 0$

Combine $x^{2}$ and $\frac{3}{3}$ .

$\frac{x^{2} \cdot 3}{3} + \frac{- 7 x \cdot 3 - 2 x}{3} + \frac{14}{3} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 - 7 x \cdot 3 - 2 x}{3} + \frac{14}{3} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 - 7 x \cdot 3 - 2 x + 14}{3} = 0$

Simplify the numerator.

Move $3$ to the left of $x^{2}$ .

$\frac{3 \cdot x^{2} - 7 x \cdot 3 - 2 x + 14}{3} = 0$

Multiply $3$ by $- 7$ .

$\frac{3 x^{2} - 21 x - 2 x + 14}{3} = 0$

Subtract $2 x$ from $- 21 x$ .

$\frac{3 x^{2} - 23 x + 14}{3} = 0$

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 $a \cdot c = 3 \cdot 14 = 42$ and whose sum is $b = - 23$ .

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

$\frac{3 x^{2} - 23 x + 14}{3} = 0$

Rewrite $- 23$ as $- 2$ plus $- 21$

$\frac{3 x^{2} + (- 2 - 21) x + 14}{3} = 0$

Apply the distributive property.

$\frac{3 x^{2} - 2 x - 21 x + 14}{3} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} - 2 x) - 21 x + 14}{3} = 0$

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

$\frac{x (3 x - 2) - 7 (3 x - 2)}{3} = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{(3 x - 2) (x - 7)}{3} = 0$

Expand $(3 x - 2) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 x (x - 7) - 2 (x - 7)}{3} = 0$

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 7 - 2 (x - 7)}{3} = 0$

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 7 - 2 x - 2 \cdot - 7}{3} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot - 7 - 2 x - 2 \cdot - 7}{3} = 0$

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot - 7 - 2 x - 2 \cdot - 7}{3} = 0$

Multiply $- 7$ by $3$ .

$\frac{3 x^{2} - 21 x - 2 x - 2 \cdot - 7}{3} = 0$

Multiply $- 2$ by $- 7$ .

$\frac{3 x^{2} - 21 x - 2 x + 14}{3} = 0$

Subtract $2 x$ from $- 21 x$ .

$\frac{3 x^{2} - 23 x + 14}{3} = 0$

Split the fraction $\frac{3 x^{2} - 23 x + 14}{3}$ into two fractions.

$\frac{3 x^{2} - 23 x}{3} + \frac{14}{3} = 0$

Split the fraction $\frac{3 x^{2} - 23 x}{3}$ into two fractions.

$\frac{3 x^{2}}{3} + \frac{- 23 x}{3} + \frac{14}{3} = 0$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- 23 x}{3} + \frac{14}{3} = 0$

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

$x^{2} + \frac{- 23 x}{3} + \frac{14}{3} = 0$

Move the negative in front of the fraction.

$x^{2} - \frac{23 x}{3} + \frac{14}{3} = 0$

The standard quadratic equation using the given set of solutions ${\frac{2}{3}, 7}$ is $y = x^{2} - \frac{23 x}{3} + \frac{14}{3}$ .

$y = x^{2} - \frac{23 x}{3} + \frac{14}{3}$

Find the Quadratic Equation Given the Roots 2/3 , 7

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