Question

Write a quadratic equation with the given roots. Write the equation in the form ax2 + bx + c = 0 where a, b, and c are integers. -10 and -2

Answer 1

A quadratic equation with roots -10 and -2 is x² + 12x + 20 = 0, derived by using the formula a(x - r1)(x - r2) = 0 and expanding it after substituting the given roots.

Step-by-step explanation:

To write a quadratic equation with the given roots -10 and -2, you can use the fact that a quadratic equation ax² + bx + c = 0 with roots r1 and r2 can be written as a(x - r1)(x - r2) = 0. Therefore, substituting the given roots into this formula, we get:

a(x + 10)(x + 2) = 0

Expanding this, it becomes:

a(x² + 2x + 10x + 20) = 0

We simplify to get:

ax² + 12ax + 20a = 0

To have the equation in standard form with integer coefficients, we can choose a = 1, so the equation becomes:

x² + 12x + 20 = 0

Answer 2

A quadratic with roots at 8 and 5 is f(x) = x^2 +12x + 20

In order to find an equation given roots you can create statements that equal 0 in order to create parenthesis. For instance we know x = 8 at one point. So, we can solve that to equal 0.

x = -2 ----> add 2 to both sides

x + 2 = 0

We can do the same for the other zero.

x = -10 ----> add 10 to both sides

x + 10 = 0

Now that we have both of these, we can multiply these two things together. This will give us the function we need.

f(x) = (x + 2)(x + 10)

f(x) = x^2 + 10x + 2x + 20

f(x) = x^2 + 12x + 20