To find the factors of the quadratic equation \(x^2 - 10x + 9 = 0\), we can factorize it as follows:

1. Start with the quadratic equation:
   \[
   x^2 - 10x + 9 = 0
   \]

2. We need to find two numbers that multiply to give the constant term \(+9\) and add up to the coefficient of the middle term \(-10\).

3. Consider the factors of \(9\) that add up to \(-10\). The pair of numbers that work are \(-1\) and \(-9\) because:
   \[
   (-1) \times (-9) = 9
   \]
   \[
   (-1) + (-9) = -10
   \]

4. Now, rewrite the middle term \(-10x\) using \(-1x\) and \(-9x\):
   \[
   x^2 - x - 9x + 9 = 0
   \]

5. Factor by grouping:
   \[
   x(x - 1) - 9(x - 1) = 0
   \]

6. Factor out the common binomial \((x - 1)\):
   \[
   (x - 1)(x - 9) = 0
   \]

So, the factors of the quadratic equation \(x^2 - 10x + 9 = 0\) are:
\[
(x - 1)(x - 9) = 0
\]

If you solve these factors, the solutions for \(x\) are:
\[
x = 1 \quad \text{and} \quad x = 9
\]

Would you like any more details or have any questions?

Here are five related questions for further practice:

1. What are the factors of \(x^2 + 7x + 12 = 0\)?
2. How do you factor \(x^2 - 12x + 35 = 0\)?
3. What are the roots of \(x^2 - 3x - 10 = 0\)?
4. How would you factor \(x^2 + 5x - 24 = 0\)?
5. Can you factor \(x^2 - 15x + 56 = 0\) and find the roots?

**Tip:** When factoring quadratic equations, always check if the middle term can be split into two numbers that multiply to the constant term and add to the coefficient of the middle term.

Learn how to factorize the quadratic equation x^2 - 10x + 9 step-by-step with detailed explanations. This problem covers key algebraic concepts, including factoring quadratic equations. Suitable for students in Grades 7-9.

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