To find the quadratic equation with roots using the formula $x^2 - (\text{sum of the roots})x + (\text{product of the roots}) = 0$, let's break down the problem step by step for the roots $-9$ and $0$.

Steps:

1. Find the sum of the roots: $\text{Sum of roots} = -9 + 0 = -9$

2. Find the product of the roots: $\text{Product of roots} = -9 \times 0 = 0$

3. Substitute the sum and product into the formula $x^2 - (\text{sum})x + (\text{product}) = 0$:

   Now, substitute the values into the formula: $x^2 - (\text{sum of the roots})x + (\text{product of the roots}) = 0$ $x^2 - (-9)x + 0 = 0$

4. Simplify the equation: $x^2 + 9x = 0$

Thus, the quadratic equation with roots $-9$ and $0$ is: $x^2 + 9x = 0$

This matches the equation we found earlier. The formula $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ provides a straightforward way to generate the quadratic equation when the roots are known.

Would you like to go through another example or discuss further details?

Related Questions:

1. How does factoring work when one of the roots is zero?
2. What is the relation between the discriminant and the type of roots in a quadratic equation?
3. How can you graph a quadratic equation based on its roots?
4. How do you derive the quadratic formula from the standard form of a quadratic equation?
5. Can you solve a quadratic equation by completing the square?
6. What happens when both roots are the same?
7. What is the axis of symmetry for a quadratic equation with known roots?
8. How do you solve a quadratic equation when the roots are complex numbers?

Tip:

When one of the roots is zero, the quadratic equation will always be factorable as $x(x + r) = 0$, where $r$ is the non-zero root.