Find the Quadratic Equation 5 , -14/3

$5$ , $- \frac{14}{3}$

Step 1

$x = 5$ and $x = - \frac{14}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - 5$ and $x + \frac{14}{3}$ are the factors of the quadratic equation.

$(x - 5) (x + \frac{14}{3}) = 0$

Step 2

Expand $(x - 5) (x + \frac{14}{3})$ using the FOIL Method.

Apply the distributive property.

$x (x + \frac{14}{3}) - 5 (x + \frac{14}{3}) = 0$

Apply the distributive property.

$x \cdot x + x (\frac{14}{3}) - 5 (x + \frac{14}{3}) = 0$

Apply the distributive property.

$x \cdot x + x (\frac{14}{3}) - 5 x - 5 (\frac{14}{3}) = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (\frac{14}{3}) - 5 x - 5 (\frac{14}{3}) = 0$

Combine $x$ and $\frac{14}{3}$ .

$x^{2} + \frac{x \cdot 14}{3} - 5 x - 5 (\frac{14}{3}) = 0$

Move $14$ to the left of $x$ .

$x^{2} + \frac{14 \cdot x}{3} - 5 x - 5 (\frac{14}{3}) = 0$

Multiply $- 5 (\frac{14}{3})$ .

Combine $- 5$ and $\frac{14}{3}$ .

$x^{2} + \frac{14 x}{3} - 5 x + \frac{- 5 \cdot 14}{3} = 0$

Multiply $- 5$ by $14$ .

$x^{2} + \frac{14 x}{3} - 5 x + \frac{- 70}{3} = 0$

Move the negative in front of the fraction.

$x^{2} + \frac{14 x}{3} - 5 x - \frac{70}{3} = 0$

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

$x^{2} + \frac{14 x}{3} - 5 x \cdot \frac{3}{3} - \frac{70}{3} = 0$

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

$x^{2} + \frac{14 x}{3} + \frac{- 5 x \cdot 3}{3} - \frac{70}{3} = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{14 x - 5 x \cdot 3}{3} - \frac{70}{3} = 0$

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

$x^{2} \cdot \frac{3}{3} + \frac{14 x - 5 x \cdot 3}{3} - \frac{70}{3} = 0$

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

$\frac{x^{2} \cdot 3}{3} + \frac{14 x - 5 x \cdot 3}{3} - \frac{70}{3} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 + 14 x - 5 x \cdot 3}{3} - \frac{70}{3} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 + 14 x - 5 x \cdot 3 - 70}{3} = 0$

Step 4

Simplify the numerator.

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

$\frac{3 \cdot x^{2} + 14 x - 5 x \cdot 3 - 70}{3} = 0$

Multiply $3$ by $- 5$ .

$\frac{3 x^{2} + 14 x - 15 x - 70}{3} = 0$

Subtract $15 x$ from $14 x$ .

$\frac{3 x^{2} - x - 70}{3} = 0$

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 $a \cdot c = 3 \cdot - 70 = - 210$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$\frac{3 x^{2} - (x) - 70}{3} = 0$

Rewrite $- 1$ as $14$ plus $- 15$

$\frac{3 x^{2} + (14 - 15) x - 70}{3} = 0$

Apply the distributive property.

$\frac{3 x^{2} + 14 x - 15 x - 70}{3} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 14 x) - 15 x - 70}{3} = 0$

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

$\frac{x (3 x + 14) - 5 (3 x + 14)}{3} = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 14$ .

$\frac{(3 x + 14) (x - 5)}{3} = 0$

Step 5

Expand $(3 x + 14) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 x (x - 5) + 14 (x - 5)}{3} = 0$

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 5 + 14 (x - 5)}{3} = 0$

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 5 + 14 x + 14 \cdot - 5}{3} = 0$

Step 6

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot - 5 + 14 x + 14 \cdot - 5}{3} = 0$

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot - 5 + 14 x + 14 \cdot - 5}{3} = 0$

Multiply $- 5$ by $3$ .

$\frac{3 x^{2} - 15 x + 14 x + 14 \cdot - 5}{3} = 0$

Multiply $14$ by $- 5$ .

$\frac{3 x^{2} - 15 x + 14 x - 70}{3} = 0$

Add $- 15 x$ and $14 x$ .

$\frac{3 x^{2} - x - 70}{3} = 0$

Step 7

Split the fraction $\frac{3 x^{2} - x - 70}{3}$ into two fractions.

$\frac{3 x^{2} - x}{3} + \frac{- 70}{3} = 0$

Step 8

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

$\frac{3 x^{2}}{3} + \frac{- x}{3} + \frac{- 70}{3} = 0$

Step 9

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- x}{3} + \frac{- 70}{3} = 0$

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

$x^{2} + \frac{- x}{3} + \frac{- 70}{3} = 0$

Step 10

Move the negative in front of the fraction.

$x^{2} - \frac{x}{3} + \frac{- 70}{3} = 0$

Step 11

Move the negative in front of the fraction.

$x^{2} - \frac{x}{3} - \frac{70}{3} = 0$

Step 12

The standard quadratic equation using the given set of solutions ${5, - \frac{14}{3}}$ is $y = x^{2} - \frac{x}{3} - \frac{70}{3}$ .

$y = x^{2} - \frac{x}{3} - \frac{70}{3}$

Step 13

Find the Quadratic Equation 5 , -14/3

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