Completing the Square | Differentiated Questions Worksheet by ... - Free Printable

To solve the problems involving completing the square, we need to rewrite each quadratic expression in the form \( (x + a)^2 + b \). Here's how to approach each problem step by step:

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Problem 1: Express in completed square form

#### 1. \( x^2 + 6x + 9 \)
- The given expression is already a perfect square trinomial.
- It can be written as:
\[
x^2 + 6x + 9 = (x + 3)^2
\]
- Answer: \( (x + 3)^2 \)

#### 2. \( x^2 + 4x + 4 \)
- The given expression is also a perfect square trinomial.
- It can be written as:
\[
x^2 + 4x + 4 = (x + 2)^2
\]
- Answer: \( (x + 2)^2 \)

#### 3. \( x^2 + 12x + 36 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 + 12x + 36 = (x + 6)^2
\]
- Answer: \( (x + 6)^2 \)

#### 4. \( x^2 - 8x + 16 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 - 8x + 16 = (x - 4)^2
\]
- Answer: \( (x - 4)^2 \)

#### 5. \( x^2 - 12x + 36 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 - 12x + 36 = (x - 6)^2
\]
- Answer: \( (x - 6)^2 \)

#### 6. \( x^2 + 18x + 81 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 + 18x + 81 = (x + 9)^2
\]
- Answer: \( (x + 9)^2 \)

#### 7. \( x^2 - 20x + 100 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 - 20x + 100 = (x - 10)^2
\]
- Answer: \( (x - 10)^2 \)

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Problem 2: Express in completed square form

#### 1. \( x^2 + 2x + 4 \)
- To complete the square:
1. Take the coefficient of \( x \), which is 2, divide it by 2, and square it: \( \left(\frac{2}{2}\right)^2 = 1 \).
2. Add and subtract this square inside the expression:
\[
x^2 + 2x + 4 = (x^2 + 2x + 1) + 3 = (x + 1)^2 + 3
\]
- Answer: \( (x + 1)^2 + 3 \)

#### 2. \( x^2 + 4x + 5 \)
- To complete the square:
1. Take the coefficient of \( x \), which is 4, divide it by 2, and square it: \( \left(\frac{4}{2}\right)^2 = 4 \).
2. Add and subtract this square inside the expression:
\[
x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1
\]
- Answer: \( (x + 2)^2 + 1 \)

#### 3. \( x^2 - 8x + 17 \)
- To complete the square:
1. Take the coefficient of \( x \), which is -8, divide it by 2, and square it: \( \left(\frac{-8}{2}\right)^2 = 16 \).
2. Add and subtract this square inside the expression:
\[
x^2 - 8x + 17 = (x^2 - 8x + 16) + 1 = (x - 4)^2 + 1
\]
- Answer: \( (x - 4)^2 + 1 \)

#### 4. \( x^2 - 6x + 12 \)
- To complete the square:
1. Take the coefficient of \( x \), which is -6, divide it by 2, and square it: \( \left(\frac{-6}{2}\right)^2 = 9 \).
2. Add and subtract this square inside the expression:
\[
x^2 - 6x + 12 = (x^2 - 6x + 9) + 3 = (x - 3)^2 + 3
\]
- Answer: \( (x - 3)^2 + 3 \)

#### 5. \( x^2 + 4x - 1 \)
- To complete the square:
1. Take the coefficient of \( x \), which is 4, divide it by 2, and square it: \( \left(\frac{4}{2}\right)^2 = 4 \).
2. Add and subtract this square inside the expression:
\[
x^2 + 4x - 1 = (x^2 + 4x + 4) - 5 = (x + 2)^2 - 5
\]
- Answer: \( (x + 2)^2 - 5 \)

#### 6. \( x^2 - 10x + 25 \)
- The given expression is a perfect square trinomial.
- It can be written as:
\[
x^2 - 10x + 25 = (x - 5)^2
\]
- Answer: \( (x - 5)^2 \)

#### 7. \( x^2 + 16x + 40 \)
- To complete the square:
1. Take the coefficient of \( x \), which is 16, divide it by 2, and square it: \( \left(\frac{16}{2}\right)^2 = 64 \).
2. Add and subtract this square inside the expression:
\[
x^2 + 16x + 40 = (x^2 + 16x + 64) - 24 = (x + 8)^2 - 24
\]
- Answer: \( (x + 8)^2 - 24 \)

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Problem 3: Express in completed square form

#### 1. \( 2x^2 + 12x - 3 \)
- Factor out the coefficient of \( x^2 \) (which is 2) from the first two terms:
\[
2x^2 + 12x - 3 = 2(x^2 + 6x) - 3
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is 6, divide it by 2, and square it: \( \left(\frac{6}{2}\right)^2 = 9 \).
2. Add and subtract this square inside the parentheses:
\[
2(x^2 + 6x) - 3 = 2(x^2 + 6x + 9 - 9) - 3 = 2((x + 3)^2 - 9) - 3
\]
- Simplify:
\[
2((x + 3)^2 - 9) - 3 = 2(x + 3)^2 - 18 - 3 = 2(x + 3)^2 - 21
\]
- Answer: \( 2(x + 3)^2 - 21 \)

#### 2. \( 2x^2 - 8x + 5 \)
- Factor out the coefficient of \( x^2 \) (which is 2) from the first two terms:
\[
2x^2 - 8x + 5 = 2(x^2 - 4x) + 5
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is -4, divide it by 2, and square it: \( \left(\frac{-4}{2}\right)^2 = 4 \).
2. Add and subtract this square inside the parentheses:
\[
2(x^2 - 4x) + 5 = 2(x^2 - 4x + 4 - 4) + 5 = 2((x - 2)^2 - 4) + 5
\]
- Simplify:
\[
2((x - 2)^2 - 4) + 5 = 2(x - 2)^2 - 8 + 5 = 2(x - 2)^2 - 3
\]
- Answer: \( 2(x - 2)^2 - 3 \)

#### 3. \( 3x^2 + 3x + 1 \)
- Factor out the coefficient of \( x^2 \) (which is 3) from the first two terms:
\[
3x^2 + 3x + 1 = 3(x^2 + x) + 1
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is 1, divide it by 2, and square it: \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \).
2. Add and subtract this square inside the parentheses:
\[
3(x^2 + x) + 1 = 3\left(x^2 + x + \frac{1}{4} - \frac{1}{4}\right) + 1 = 3\left(\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 1
\]
- Simplify:
\[
3\left(\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 1 = 3\left(x + \frac{1}{2}\right)^2 - \frac{3}{4} + 1 = 3\left(x + \frac{1}{2}\right)^2 + \frac{1}{4}
\]
- Answer: \( 3\left(x + \frac{1}{2}\right)^2 + \frac{1}{4} \)

#### 4. \( 4x^2 + 2x + 1 \)
- Factor out the coefficient of \( x^2 \) (which is 4) from the first two terms:
\[
4x^2 + 2x + 1 = 4(x^2 + \frac{1}{2}x) + 1
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is \( \frac{1}{2} \), divide it by 2, and square it: \( \left(\frac{\frac{1}{2}}{2}\right)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \).
2. Add and subtract this square inside the parentheses:
\[
4(x^2 + \frac{1}{2}x) + 1 = 4\left(x^2 + \frac{1}{2}x + \frac{1}{16} - \frac{1}{16}\right) + 1 = 4\left(\left(x + \frac{1}{4}\right)^2 - \frac{1}{16}\right) + 1
\]
- Simplify:
\[
4\left(\left(x + \frac{1}{4}\right)^2 - \frac{1}{16}\right) + 1 = 4\left(x + \frac{1}{4}\right)^2 - \frac{4}{16} + 1 = 4\left(x + \frac{1}{4}\right)^2 - \frac{1}{4} + 1 = 4\left(x + \frac{1}{4}\right)^2 + \frac{3}{4}
\]
- Answer: \( 4\left(x + \frac{1}{4}\right)^2 + \frac{3}{4} \)

#### 5. \( 5x^2 - 5x + 2 \)
- Factor out the coefficient of \( x^2 \) (which is 5) from the first two terms:
\[
5x^2 - 5x + 2 = 5(x^2 - x) + 2
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is -1, divide it by 2, and square it: \( \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \).
2. Add and subtract this square inside the parentheses:
\[
5(x^2 - x) + 2 = 5\left(x^2 - x + \frac{1}{4} - \frac{1}{4}\right) + 2 = 5\left(\left(x - \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 2
\]
- Simplify:
\[
5\left(\left(x - \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 2 = 5\left(x - \frac{1}{2}\right)^2 - \frac{5}{4} + 2 = 5\left(x - \frac{1}{2}\right)^2 + \frac{3}{4}
\]
- Answer: \( 5\left(x - \frac{1}{2}\right)^2 + \frac{3}{4} \)

#### 6. \( 2x^2 + 16x + 40 \)
- Factor out the coefficient of \( x^2 \) (which is 2) from the first two terms:
\[
2x^2 + 16x + 40 = 2(x^2 + 8x) + 40
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is 8, divide it by 2, and square it: \( \left(\frac{8}{2}\right)^2 = 16 \).
2. Add and subtract this square inside the parentheses:
\[
2(x^2 + 8x) + 40 = 2(x^2 + 8x + 16 - 16) + 40 = 2((x + 4)^2 - 16) + 40
\]
- Simplify:
\[
2((x + 4)^2 - 16) + 40 = 2(x + 4)^2 - 32 + 40 = 2(x + 4)^2 + 8
\]
- Answer: \( 2(x + 4)^2 + 8 \)

#### 7. \( 4x^2 + 8x - 5 \)
- Factor out the coefficient of \( x^2 \) (which is 4) from the first two terms:
\[
4x^2 + 8x - 5 = 4(x^2 + 2x) - 5
\]
- Complete the square inside the parentheses:
1. Take the coefficient of \( x \), which is 2, divide it by 2, and square it: \( \left(\frac{2}{2}\right)^2 = 1 \).
2. Add and subtract this square inside the parentheses:
\[
4(x^2 + 2x) - 5 = 4(x^2 + 2x + 1 - 1) - 5 = 4((x + 1)^2 - 1) - 5
\]
- Simplify:
\[
4((x + 1)^2 - 1) - 5 = 4(x + 1)^2 - 4 - 5 = 4(x + 1)^2 - 9
\]
- Answer: \( 4(x + 1)^2 - 9 \)

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Final Answer

\[
\boxed{
\begin{array}{ll}
\text{Problem 1:} & (x + 3)^2, (x + 2)^2, (x + 6)^2, (x - 4)^2, (x - 6)^2, (x + 9)^2, (x - 10)^2 \\
\text{Problem 2:} & (x + 1)^2 + 3, (x + 2)^2 + 1, (x - 4)^2 + 1, (x - 3)^2 + 3, (x + 2)^2 - 5, (x - 5)^2, (x + 8)^2 - 24 \\
\text{Problem 3:} & 2(x + 3)^2 - 21, 2(x - 2)^2 - 3, 3\left(x + \frac{1}{2}\right)^2 + \frac{1}{4}, 4\left(x + \frac{1}{4}\right)^2 + \frac{3}{4}, 5\left(x - \frac{1}{2}\right)^2 + \frac{3}{4}, 2(x + 4)^2 + 8, 4(x + 1)^2 - 9
\end{array}
}
\]