Find the Quadratic Equation Given the Roots -2 and 3

$- 2$ and $3$

Step 1

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

$(x + 2) (x - 3) = 0$

Step 2

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

Apply the distributive property.

$x (x - 3) + 2 (x - 3) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 2 (x - 3) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3 = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $2$ by $- 3$ .

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

Add $- 3 x$ and $2 x$ .

$x^{2} - x - 6 = 0$

Step 4

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

$y = x^{2} - x - 6$

Step 5

Find the Quadratic Equation Given the Roots -2 and 3

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