Solving square root equations | TPT - Free Printable

Problem Overview:

The task involves solving quadratic equations using the square root method. Each equation is solved, and the solutions are used to decode a hidden message by matching them with corresponding letters.

Steps to Solve:

1. Understand the Square Root Method:
- The square root method is used when the quadratic equation is in the form \( (x + a)^2 = b \).
- To solve, take the square root of both sides: \( x + a = \pm \sqrt{b} \).
- Then, isolate \( x \) by subtracting \( a \) from both sides: \( x = -a \pm \sqrt{b} \).

2. Solve Each Equation:
- For each equation, simplify it to the form \( (x + a)^2 = b \), then apply the square root method.
- Record the solutions and match them with the corresponding letters.

3. Decode the Message:
- Use the solutions to determine the correct order of letters, which will reveal the hidden message.

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Solving Each Equation:

#### 1. \( (x - 4)^2 = 25 \)
- Take the square root of both sides:
\[
x - 4 = \pm \sqrt{25}
\]
\[
x - 4 = \pm 5
\]
- Solve for \( x \):
\[
x = 4 + 5 \quad \text{or} \quad x = 4 - 5
\]
\[
x = 9 \quad \text{or} \quad x = -1
\]
- Solutions: \( (9, -1) \)
- Match with letter: T

#### 2. \( 7(x + 3)^2 - 3 = 60 \)
- Isolate \( (x + 3)^2 \):
\[
7(x + 3)^2 = 63
\]
\[
(x + 3)^2 = 9
\]
- Take the square root of both sides:
\[
x + 3 = \pm \sqrt{9}
\]
\[
x + 3 = \pm 3
\]
- Solve for \( x \):
\[
x = -3 + 3 \quad \text{or} \quad x = -3 - 3
\]
\[
x = 0 \quad \text{or} \quad x = -6
\]
- Solutions: \( (0, -6) \)
- Match with letter: L

#### 3. \( D \): \( 7(x + 3)^2 - 3 = 60 \)
- This is the same as the previous equation, so the solutions are:
\[
(0, -6)
\]
- Match with letter: L

#### 4. \( Q \): \( (x + 1)^2 + 4 = 8 \)
- Isolate \( (x + 1)^2 \):
\[
(x + 1)^2 = 4
\]
- Take the square root of both sides:
\[
x + 1 = \pm \sqrt{4}
\]
\[
x + 1 = \pm 2
\]
- Solve for \( x \):
\[
x = -1 + 2 \quad \text{or} \quad x = -1 - 2
\]
\[
x = 1 \quad \text{or} \quad x = -3
\]
- Solutions: \( (1, -3) \)
- Match with letter: Q

#### 5. \( U \): \( 2(x - 3)^2 + 2 = 74 \)
- Isolate \( (x - 3)^2 \):
\[
2(x - 3)^2 = 72
\]
\[
(x - 3)^2 = 36
\]
- Take the square root of both sides:
\[
x - 3 = \pm \sqrt{36}
\]
\[
x - 3 = \pm 6
\]
- Solve for \( x \):
\[
x = 3 + 6 \quad \text{or} \quad x = 3 - 6
\]
\[
x = 9 \quad \text{or} \quad x = -3
\]
- Solutions: \( (9, -3) \)
- Match with letter: U

#### 6. \( E \): \( (x + 5)^2 - 2 = 98 \)
- Isolate \( (x + 5)^2 \):
\[
(x + 5)^2 = 100
\]
- Take the square root of both sides:
\[
x + 5 = \pm \sqrt{100}
\]
\[
x + 5 = \pm 10
\]
- Solve for \( x \):
\[
x = -5 + 10 \quad \text{or} \quad x = -5 - 10
\]
\[
x = 5 \quad \text{or} \quad x = -15
\]
- Solutions: \( (5, -15) \)
- Match with letter: E

#### 7. \( O \): \( 4(x - 2)^2 = 64 \)
- Isolate \( (x - 2)^2 \):
\[
(x - 2)^2 = 16
\]
- Take the square root of both sides:
\[
x - 2 = \pm \sqrt{16}
\]
\[
x - 2 = \pm 4
\]
- Solve for \( x \):
\[
x = 2 + 4 \quad \text{or} \quad x = 2 - 4
\]
\[
x = 6 \quad \text{or} \quad x = -2
\]
- Solutions: \( (6, -2) \)
- Match with letter: O

#### 8. \( T \): \( 8(x - 2)^2 + 2 = 10 \)
- Isolate \( (x - 2)^2 \):
\[
8(x - 2)^2 = 8
\]
\[
(x - 2)^2 = 1
\]
- Take the square root of both sides:
\[
x - 2 = \pm \sqrt{1}
\]
\[
x - 2 = \pm 1
\]
- Solve for \( x \):
\[
x = 2 + 1 \quad \text{or} \quad x = 2 - 1
\]
\[
x = 3 \quad \text{or} \quad x = 1
\]
- Solutions: \( (3, 1) \)
- Match with letter: T

#### 9. \( B \): \( (x + 7)^2 - 6 = 58 \)
- Isolate \( (x + 7)^2 \):
\[
(x + 7)^2 = 64
\]
- Take the square root of both sides:
\[
x + 7 = \pm \sqrt{64}
\]
\[
x + 7 = \pm 8
\]
- Solve for \( x \):
\[
x = -7 + 8 \quad \text{or} \quad x = -7 - 8
\]
\[
x = 1 \quad \text{or} \quad x = -15
\]
- Solutions: \( (1, -15) \)
- Match with letter: B

#### 10. \( S \): \( -3(x - 1)^2 = -12 \)
- Isolate \( (x - 1)^2 \):
\[
(x - 1)^2 = 4
\]
- Take the square root of both sides:
\[
x - 1 = \pm \sqrt{4}
\]
\[
x - 1 = \pm 2
\]
- Solve for \( x \):
\[
x = 1 + 2 \quad \text{or} \quad x = 1 - 2
\]
\[
x = 3 \quad \text{or} \quad x = -1
\]
- Solutions: \( (3, -1) \)
- Match with letter: S

---

Decoding the Message:

The solutions correspond to the following letters:
- \( (9, -1) \rightarrow T \)
- \( (0, -6) \rightarrow L \)
- \( (1, -3) \rightarrow Q \)
- \( (9, -3) \rightarrow U \)
- \( (5, -15) \rightarrow E \)
- \( (6, -2) \rightarrow O \)
- \( (3, 1) \rightarrow T \)
- \( (1, -15) \rightarrow B \)
- \( (3, -1) \rightarrow S \)

The hidden message is: TOLQUETBS

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Final Answer:

\[
\boxed{TOLQUETBS}
\]