Find the Quadratic Equation Given the Roots 5 and 7

$5$ and $7$

Step 1

$x = 5$ and $x = 7$ are the two real distinct solutions for the quadratic equation, which means that $x - 5$ and $x - 7$ are the factors of the quadratic equation.

$(x - 5) (x - 7) = 0$

Step 2

Expand $(x - 5) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$x (x - 7) - 5 (x - 7) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - 5 (x - 7) = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 7 - 5 x - 5 \cdot - 7 = 0$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 7 - 5 x - 5 \cdot - 7 = 0$

Move $- 7$ to the left of $x$ .

$x^{2} - 7 \cdot x - 5 x - 5 \cdot - 7 = 0$

Multiply $- 5$ by $- 7$ .

$x^{2} - 7 x - 5 x + 35 = 0$

Subtract $5 x$ from $- 7 x$ .

$x^{2} - 12 x + 35 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${5, 7}$ is $y = x^{2} - 12 x + 35$ .

$y = x^{2} - 12 x + 35$

Step 5

Find the Quadratic Equation Given the Roots 5 and 7

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