Find the number to add to  to make it a perfect square trinomial.

To find the number to add to an expression to make it a perfect square trinomial, let's start with the general form of a quadratic expression:

\[
ax^2 + bx + c
\]

A perfect square trinomial has the form:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Given a quadratic expression like \(x^2 + bx\), we need to add a term to complete the square and make it a perfect square trinomial.

### Step-by-Step Solution

1. **Identify the given expression:** The expression is usually in the form \(x^2 + bx\).
   
2. **Find half of the coefficient of \(x\):** Take the coefficient of \(x\) (which is \(b\)) and divide it by 2.

   \[
   \frac{b}{2}
   \]

3. **Square this value:** Square the result of the previous step to get the term that completes the square.

   \[
   \left(\frac{b}{2}\right)^2 = \frac{b^2}{4}
   \]

4. **Add this term to make the expression a perfect square trinomial:** Add \(\frac{b^2}{4}\) to the expression \(x^2 + bx\).

### Example:

For the expression \(x^2 + 6x\):

1. The coefficient of \(x\) is 6.
2. Half of 6 is \( \frac{6}{2} = 3 \).
3. Squaring 3 gives \(3^2 = 9\).

Thus, adding 9 makes the expression a perfect square trinomial:

\[
x^2 + 6x + 9 = (x + 3)^2
\]

Would you like further details or have any questions?

### Expanding Further:
1. What is a perfect square trinomial?
2. How do you factor perfect square trinomials?
3. How to complete the square for a general quadratic equation?
4. Why is completing the square useful in solving equations?
5. Can this method be applied to any quadratic expression?

**Tip:** Completing the square is also a key step in deriving the quadratic formula!

Learn how to find the number to add to a quadratic expression to complete the square and form a perfect square trinomial. This detailed explanation covers completing the square technique, suitable for algebra students in Grades 7-9.

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