Express in the form of (x-m)^2+n to the equation x^2+9x

To express the quadratic expression \(x^2 + 9x\) in the form \((x - m)^2 + n\), we need to complete the square. Here are the steps:

1. **Rewrite the expression**: Start with the given expression.
   \[
   x^2 + 9x
   \]

2. **Complete the square**: Add and subtract a constant to make a perfect square trinomial.
   \[
   x^2 + 9x + \left(\frac{9}{2}\right)^2 - \left(\frac{9}{2}\right)^2
   \]
   Here, \(\left(\frac{9}{2}\right)^2 = \frac{81}{4}\).

3. **Form the perfect square trinomial**: Group the first three terms.
   \[
   \left(x^2 + 9x + \frac{81}{4}\right) - \frac{81}{4}
   \]

4. **Factor the trinomial**: The trinomial inside the parentheses is a perfect square.
   \[
   \left(x + \frac{9}{2}\right)^2 - \frac{81}{4}
   \]

5. **Simplify the expression**: We now have the quadratic expression in the form \((x - m)^2 + n\).
   \[
   \left(x - \left(-\frac{9}{2}\right)\right)^2 - \frac{81}{4}
   \]

So, the expression \(x^2 + 9x\) in the form \((x - m)^2 + n\) is:
\[
(x - (-\frac{9}{2}))^2 - \frac{81}{4}
\]
or more simply:
\[
(x + \frac{9}{2})^2 - \frac{81}{4}
\]

Would you like more details on this process or have any other questions?

Here are some more questions you might want to explore:
1. How do you complete the square for a quadratic expression with a coefficient other than 1 for \(x^2\)?
2. Can you show how to solve a quadratic equation by completing the square?
3. What is the vertex form of a quadratic function?
4. How does completing the square help in finding the vertex of a parabola?
5. Can you derive the quadratic formula using the method of completing the square?

**Tip:** When completing the square, the constant term you add inside the parentheses is \(\left(\frac{b}{2}\right)^2\), where \(b\) is the coefficient of \(x\). This helps to form a perfect square trinomial.

Learn how to express the quadratic expression x^2 + 9x in the form (x - m)^2 + n using the method of completing the square. This problem covers key algebraic concepts and the quadratic formula. Suitable for students in Grades 10-12.

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