Find the Quadratic Equation Given the Roots 5 and -2

$5$ and $- 2$

Step 1

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

$(x - 5) (x + 2) = 0$

Step 2

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

Apply the distributive property.

$x (x + 2) - 5 (x + 2) = 0$

Apply the distributive property.

$x \cdot x + x \cdot 2 - 5 (x + 2) = 0$

Apply the distributive property.

$x \cdot x + x \cdot 2 - 5 x - 5 \cdot 2 = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 5$ by $2$ .

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

Subtract $5 x$ from $2 x$ .

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

Step 4

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

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

Step 5

Find the Quadratic Equation Given the Roots 5 and -2

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