Find the Quadratic Equation 2 , -5/3

$2$ , $- \frac{5}{3}$

Step 1

$x = 2$ and $x = - \frac{5}{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - 2$ and $x + \frac{5}{3}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - 2) (x + \frac{5}{3})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $- 2 (\frac{5}{3})$ .

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

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

Multiply $- 2$ by $5$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

Combine the numerators over the common denominator.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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

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

Multiply $3$ by $- 2$ .

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

Subtract $6 x$ from $5 x$ .

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

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 $a \cdot c = 3 \cdot - 10 = - 30$ and whose sum is $b = - 1$ .

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

$\frac{3 x^{2} - (x) - 10}{3} = 0$

Rewrite $- 1$ as $5$ plus $- 6$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 5$ .

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

Step 5

Expand $(3 x + 5) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $3$ .

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

Multiply $5$ by $- 2$ .

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

Add $- 6 x$ and $5 x$ .

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

Step 7

Split the fraction $\frac{3 x^{2} - x - 10}{3}$ into two fractions.

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

Step 8

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

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

Step 9

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Step 10

Move the negative in front of the fraction.

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

Find the Quadratic Equation 2 , -5/3

Related Questions

Support LCMGCF.com by disabling your adblocker.