The equation you provided is:

\[
x^2 + 2 = 9x
\]

To solve this, we will first rearrange the terms so that we have a standard quadratic equation.

1. Subtract \(9x\) from both sides:

\[
x^2 - 9x + 2 = 0
\]

Now we have a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -9\), and \(c = 2\).

We can solve this using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \(a\), \(b\), and \(c\):

\[
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(2)}}{2(1)}
\]

\[
x = \frac{9 \pm \sqrt{81 - 8}}{2}
\]

\[
x = \frac{9 \pm \sqrt{73}}{2}
\]

Thus, the two possible solutions are:

\[
x = \frac{9 + \sqrt{73}}{2} \quad \text{or} \quad x = \frac{9 - \sqrt{73}}{2}
\]

These are the exact solutions. For an approximate numerical solution:

\[
x \approx \frac{9 + 8.54}{2} \approx 8.77 \quad \text{or} \quad x \approx \frac{9 - 8.54}{2} \approx 0.23
\]

Would you like further details or clarifications on this?

Here are some related questions to expand your understanding:
1. How can you factorize quadratic equations instead of using the quadratic formula?
2. What is the discriminant and how does it help in solving quadratic equations?
3. How do you graph a quadratic equation like \(x^2 - 9x + 2 = 0\)?
4. How does the quadratic formula change for different values of \(a\), \(b\), and \(c\)?
5. Can you explain the meaning of complex roots when the discriminant is negative?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before using the quadratic formula to determine if real or complex solutions exist.

Explore how to solve the quadratic equation x^2 - 9x + 2 = 0 using the quadratic formula. This detailed step-by-step solution covers essential algebraic concepts, including how to use the quadratic formula and interpret the discriminant. Perfect for students in Grades 8-10.

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