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 $- 3$ .

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

Subtract $2 x$ from $- 3 x$ .

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

Step 4

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

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

Step 5

Find the Quadratic Equation Given the Roots 2 and 3

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