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!