To find \(a\) and \(b\) in the quadratic equation \(x^2 + ax + b = 0\), we need to know the roots of the equation. Suppose the roots of the equation are \(r_1\) and \(r_2\). The quadratic equation can be expressed in terms of its roots as follows:

\[ (x - r_1)(x - r_2) = 0 \]

Expanding this, we get:

\[ x^2 - (r_1 + r_2)x + r_1r_2 = 0 \]

By comparing this with the standard form \(x^2 + ax + b = 0\), we can identify the coefficients \(a\) and \(b\) as follows:

- \(a = -(r_1 + r_2)\)
- \(b = r_1r_2\)

So, given the roots \(r_1\) and \(r_2\), you can find \(a\) and \(b\) using these formulas.

Would you like an example to illustrate this?

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Here are five questions you might ask next:
1. If the roots of \(x^2 + ax + b = 0\) are 2 and 3, what are \(a\) and \(b\)?
2. How do you find the roots of \(x^2 + 4x + 4 = 0\)?
3. What are the coefficients \(a\) and \(b\) if the roots are -1 and -5?
4. Can you derive the quadratic equation given roots 3 and -2?
5. How does the sum and product of roots relate to the coefficients in a quadratic equation?

**Tip:** The sum and product of the roots of a quadratic equation provide a straightforward way to determine the coefficients \(a\) and \(b\).

Learn how to find coefficients a and b of the quadratic equation x^2 + ax + b = 0 using Vieta's formulas. This problem covers key algebraic concepts, including quadratic equations and their roots. Suitable for high school students in Grades 10-12.

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