Completing The Square Practice Worksheet, 50% OFF - Free Printable

Problem Analysis:

The task involves completing the square for quadratic expressions. Completing the square is a method used to rewrite a quadratic expression in the form of a perfect square trinomial plus (or minus) a constant. The general steps are:

1. Start with a quadratic expression in the form \( ax^2 + bx + c \).
2. Ensure the coefficient of \( x^2 \) is 1 (if not, divide through by \( a \)).
3. Take the coefficient of \( x \) (which is \( b \)), divide it by 2, and square it: \( \left(\frac{b}{2}\right)^2 \).
4. Add and subtract this value inside the expression to complete the square.
5. Rewrite the expression as a perfect square trinomial plus (or minus) a constant.

Let's solve each problem step by step.

---

Problem 1: \( x^2 + 8x + ? = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 8 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 8x + 16 - 16 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 4)^2 - 16 = 0
\]

Thus, the value to complete the square is \( 16 \).

Answer:
\[
\boxed{16}
\]

---

Problem 2: \( x^2 + 6x + ? = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 6 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{6}{2}\right)^2 = 3^2 = 9
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 6x + 9 - 9 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 3)^2 - 9 = 0
\]

Thus, the value to complete the square is \( 9 \).

Answer:
\[
\boxed{9}
\]

---

Problem 3: \( 2x^2 - 20x - 48 = 0 \)

#### Step 1: Factor out the coefficient of \( x^2 \) (which is 2).
\[
2(x^2 - 10x - 24) = 0
\]

#### Step 2: Focus on completing the square for \( x^2 - 10x \).
- Coefficient of \( x \) is \( -10 \).
- Compute \( \left(\frac{-10}{2}\right)^2 \):
\[
\left(\frac{-10}{2}\right)^2 = (-5)^2 = 25
\]

#### Step 3: Add and subtract this value inside the parentheses.
\[
x^2 - 10x + 25 - 25 - 24 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x - 5)^2 - 49 = 0
\]

#### Step 5: Multiply back by 2.
\[
2((x - 5)^2 - 49) = 0
\]
\[
2(x - 5)^2 - 98 = 0
\]

Thus, the value to complete the square is \( 25 \).

Answer:
\[
\boxed{25}
\]

---

Problem 4: \( x^2 + 16x + ? = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 16 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 16x + 64 - 64 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 8)^2 - 64 = 0
\]

Thus, the value to complete the square is \( 64 \).

Answer:
\[
\boxed{64}
\]

---

Problem 5: \( x^2 + 9x + 10 = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 9 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{9}{2}\right)^2 = \left(4.5\right)^2 = 20.25
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 9x + 20.25 - 20.25 + 10 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 4.5)^2 - 10.25 = 0
\]

Thus, the value to complete the square is \( 20.25 \).

Answer:
\[
\boxed{20.25}
\]

---

Problem 6: \( x^2 + 8x + 7 = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 8 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 8x + 16 - 16 + 7 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 4)^2 - 9 = 0
\]

Thus, the value to complete the square is \( 16 \).

Answer:
\[
\boxed{16}
\]

---

Problem 7: \( x^2 + 24x - 18 = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 24 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{24}{2}\right)^2 = 12^2 = 144
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 24x + 144 - 144 - 18 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 12)^2 - 162 = 0
\]

Thus, the value to complete the square is \( 144 \).

Answer:
\[
\boxed{144}
\]

---

Problem 8: \( x^2 + 45x - 3 = 0 \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 45 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{45}{2}\right)^2 = \left(22.5\right)^2 = 506.25
\]

#### Step 3: Add and subtract this value.
\[
x^2 + 45x + 506.25 - 506.25 - 3 = 0
\]

#### Step 4: Rewrite as a perfect square trinomial.
\[
(x + 22.5)^2 - 509.25 = 0
\]

Thus, the value to complete the square is \( 506.25 \).

Answer:
\[
\boxed{506.25}
\]

---

Problem 9: \( x^2 + 16x + ? \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( 16 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]

Thus, the value to complete the square is \( 64 \).

Answer:
\[
\boxed{64}
\]

---

Problem 10: \( x^2 - 2x + ? \)

#### Step 1: Identify the coefficient of \( x \).
The coefficient of \( x \) is \( -2 \).

#### Step 2: Compute \( \left(\frac{b}{2}\right)^2 \).
\[
\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1
\]

Thus, the value to complete the square is \( 1 \).

Answer:
\[
\boxed{1}
\]

---

Final Answers:

\[
\boxed{16, 9, 25, 64, 20.25, 16, 144, 506.25, 64, 1}
\]