Completing the Square math worksheet with quadratic equations and trinomial completion problems.

Explanation:
We are asked to find what number goes in the box to make each expression a *perfect square trinomial*. A perfect square trinomial looks like:

$$
x^2 + bx + c = (x + d)^2
$$

When you expand $(x + d)^2$, you get:
$$
x^2 + 2dx + d^2
$$

So, for $x^2 + bx + \boxed{?}$ to be a perfect square:
- The middle term is $bx = 2dx$ → so $d = \frac{b}{2}$
- Then the constant term must be $d^2 = \left(\frac{b}{2}\right)^2$

Let’s apply this to problems 9 and 10.

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Problem 9:
$x^2 + 16x + \boxed{?}$

Here, $b = 16$, so:
$$
d = \frac{16}{2} = 8 \\
\text{Constant term} = d^2 = 8^2 = 64
$$

✔ So the box should be 64.

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Problem 10:
$x^2 - 3x + \boxed{?}$

Here, $b = -3$, so:
$$
d = \frac{-3}{2} = -\frac{3}{2} \\
\text{Constant term} = d^2 = \left(-\frac{3}{2}\right)^2 = \frac{9}{4}
$$

✔ So the box should be $\frac{9}{4}$.

Let’s double-check both:

- $x^2 + 16x + 64 = (x + 8)^2$ ✔️
- $x^2 - 3x + \frac{9}{4} = \left(x - \frac{3}{2}\right)^2$ ✔️

Both are perfect squares.

Final Answer:
9. 64
10. $\frac{9}{4}$