Solving Quadratic Equations (A) Worksheet | Fun and Engaging ... - Free Printable

Problem: Solving Quadratic Equations

The task is to solve the given quadratic equations. Let's break it down into sections and solve each equation step by step.

---

Section A: Basic Quadratic Equations

#### 1) \( x^2 = 25 \)
To solve for \( x \), take the square root of both sides:
\[
x = \pm \sqrt{25}
\]
\[
x = \pm 5
\]
So, the solutions are:
\[
x = 5 \quad \text{and} \quad x = -5
\]

#### 2) \( x^2 = 16 \)
Similarly, take the square root of both sides:
\[
x = \pm \sqrt{16}
\]
\[
x = \pm 4
\]
So, the solutions are:
\[
x = 4 \quad \text{and} \quad x = -4
\]

#### 3) \( x^2 = 81 \)
Take the square root of both sides:
\[
x = \pm \sqrt{81}
\]
\[
x = \pm 9
\]
So, the solutions are:
\[
x = 9 \quad \text{and} \quad x = -9
\]

#### 4) \( x^2 = 100 \)
Take the square root of both sides:
\[
x = \pm \sqrt{100}
\]
\[
x = \pm 10
\]
So, the solutions are:
\[
x = 10 \quad \text{and} \quad x = -10
\]

#### 5) \( -x^2 = -64 \)
First, divide both sides by \(-1\):
\[
x^2 = 64
\]
Now, take the square root of both sides:
\[
x = \pm \sqrt{64}
\]
\[
x = \pm 8
\]
So, the solutions are:
\[
x = 8 \quad \text{and} \quad x = -8
\]

#### 6) \( -x^2 = -9 \)
Divide both sides by \(-1\):
\[
x^2 = 9
\]
Take the square root of both sides:
\[
x = \pm \sqrt{9}
\]
\[
x = \pm 3
\]
So, the solutions are:
\[
x = 3 \quad \text{and} \quad x = -3
\]

#### 7) \( -x^2 = -4 \)
Divide both sides by \(-1\):
\[
x^2 = 4
\]
Take the square root of both sides:
\[
x = \pm \sqrt{4}
\]
\[
x = \pm 2
\]
So, the solutions are:
\[
x = 2 \quad \text{and} \quad x = -2
\]

#### 8) \( -x^2 = -49 \)
Divide both sides by \(-1\):
\[
x^2 = 49
\]
Take the square root of both sides:
\[
x = \pm \sqrt{49}
\]
\[
x = \pm 7
\]
So, the solutions are:
\[
x = 7 \quad \text{and} \quad x = -7
\]

#### 9) \( x^2 = 3600 \)
Take the square root of both sides:
\[
x = \pm \sqrt{3600}
\]
\[
x = \pm 60
\]
So, the solutions are:
\[
x = 60 \quad \text{and} \quad x = -60
\]

#### 10) \( x^2 = 12100 \)
Take the square root of both sides:
\[
x = \pm \sqrt{12100}
\]
\[
x = \pm 110
\]
So, the solutions are:
\[
x = 110 \quad \text{and} \quad x = -110
\]

#### 11) \( x^2 = 14400 \)
Take the square root of both sides:
\[
x = \pm \sqrt{14400}
\]
\[
x = \pm 120
\]
So, the solutions are:
\[
x = 120 \quad \text{and} \quad x = -120
\]

#### 12) \( x^2 = 10000 \)
Take the square root of both sides:
\[
x = \pm \sqrt{10000}
\]
\[
x = \pm 100
\]
So, the solutions are:
\[
x = 100 \quad \text{and} \quad x = -100
\]

---

Section B: More Complex Quadratic Equations

#### 1) \( 2x^2 = 2 \)
Divide both sides by 2:
\[
x^2 = 1
\]
Take the square root of both sides:
\[
x = \pm \sqrt{1}
\]
\[
x = \pm 1
\]
So, the solutions are:
\[
x = 1 \quad \text{and} \quad x = -1
\]

#### 2) \( 2x^2 = 200 \)
Divide both sides by 2:
\[
x^2 = 100
\]
Take the square root of both sides:
\[
x = \pm \sqrt{100}
\]
\[
x = \pm 10
\]
So, the solutions are:
\[
x = 10 \quad \text{and} \quad x = -10
\]

#### 3) \( 2x^2 = 98 \)
Divide both sides by 2:
\[
x^2 = 49
\]
Take the square root of both sides:
\[
x = \pm \sqrt{49}
\]
\[
x = \pm 7
\]
So, the solutions are:
\[
x = 7 \quad \text{and} \quad x = -7
\]

#### 4) \( 3x^2 = 27 \)
Divide both sides by 3:
\[
x^2 = 9
\]
Take the square root of both sides:
\[
x = \pm \sqrt{9}
\]
\[
x = \pm 3
\]
So, the solutions are:
\[
x = 3 \quad \text{and} \quad x = -3
\]

#### 5) \( 3x^2 = 7500 \)
Divide both sides by 3:
\[
x^2 = 2500
\]
Take the square root of both sides:
\[
x = \pm \sqrt{2500}
\]
\[
x = \pm 50
\]
So, the solutions are:
\[
x = 50 \quad \text{and} \quad x = -50
\]

#### 6) \( 7x^2 = 2800 \)
Divide both sides by 7:
\[
x^2 = 400
\]
Take the square root of both sides:
\[
x = \pm \sqrt{400}
\]
\[
x = \pm 20
\]
So, the solutions are:
\[
x = 20 \quad \text{and} \quad x = -20
\]

#### 7) \( -2x^2 = -162 \)
Divide both sides by \(-2\):
\[
x^2 = 81
\]
Take the square root of both sides:
\[
x = \pm \sqrt{81}
\]
\[
x = \pm 9
\]
So, the solutions are:
\[
x = 9 \quad \text{and} \quad x = -9
\]

#### 8) \( -4x^2 = -256 \)
Divide both sides by \(-4\):
\[
x^2 = 64
\]
Take the square root of both sides:
\[
x = \pm \sqrt{64}
\]
\[
x = \pm 8
\]
So, the solutions are:
\[
x = 8 \quad \text{and} \quad x = -8
\]

#### 9) \( (x + 3)^2 = 16 \)
Take the square root of both sides:
\[
x + 3 = \pm \sqrt{16}
\]
\[
x + 3 = \pm 4
\]
This gives two equations:
\[
x + 3 = 4 \quad \text{or} \quad x + 3 = -4
\]
Solve each:
\[
x = 4 - 3 = 1 \quad \text{or} \quad x = -4 - 3 = -7
\]
So, the solutions are:
\[
x = 1 \quad \text{and} \quad x = -7
\]

#### 10) \( (x + 9)^2 = 36 \)
Take the square root of both sides:
\[
x + 9 = \pm \sqrt{36}
\]
\[
x + 9 = \pm 6
\]
This gives two equations:
\[
x + 9 = 6 \quad \text{or} \quad x + 9 = -6
\]
Solve each:
\[
x = 6 - 9 = -3 \quad \text{or} \quad x = -6 - 9 = -15
\]
So, the solutions are:
\[
x = -3 \quad \text{and} \quad x = -15
\]

#### 11) \( (x - 4)^2 = 81 \)
Take the square root of both sides:
\[
x - 4 = \pm \sqrt{81}
\]
\[
x - 4 = \pm 9
\]
This gives two equations:
\[
x - 4 = 9 \quad \text{or} \quad x - 4 = -9
\]
Solve each:
\[
x = 9 + 4 = 13 \quad \text{or} \quad x = -9 + 4 = -5
\]
So, the solutions are:
\[
x = 13 \quad \text{and} \quad x = -5
\]

#### 12) \( -(x - 12)^2 = -121 \)
Divide both sides by \(-1\):
\[
(x - 12)^2 = 121
\]
Take the square root of both sides:
\[
x - 12 = \pm \sqrt{121}
\]
\[
x - 12 = \pm 11
\]
This gives two equations:
\[
x - 12 = 11 \quad \text{or} \quad x - 12 = -11
\]
Solve each:
\[
x = 11 + 12 = 23 \quad \text{or} \quad x = -11 + 12 = 1
\]
So, the solutions are:
\[
x = 23 \quad \text{and} \quad x = 1
\]

---

Final Answers:

Section A:
1. \( x = 5, -5 \)
2. \( x = 4, -4 \)
3. \( x = 9, -9 \)
4. \( x = 10, -10 \)
5. \( x = 8, -8 \)
6. \( x = 3, -3 \)
7. \( x = 2, -2 \)
8. \( x = 7, -7 \)
9. \( x = 60, -60 \)
10. \( x = 110, -110 \)
11. \( x = 120, -120 \)
12. \( x = 100, -100 \)

Section B:
1. \( x = 1, -1 \)
2. \( x = 10, -10 \)
3. \( x = 7, -7 \)
4. \( x = 3, -3 \)
5. \( x = 50, -50 \)
6. \( x = 20, -20 \)
7. \( x = 9, -9 \)
8. \( x = 8, -8 \)
9. \( x = 1, -7 \)
10. \( x = -3, -15 \)
11. \( x = 13, -5 \)
12. \( x = 23, 1 \)

\boxed{
\begin{array}{ll}
\text{Section A:} & \text{Section B:} \\
1. \pm 5 & 1. \pm 1 \\
2. \pm 4 & 2. \pm 10 \\
3. \pm 9 & 3. \pm 7 \\
4. \pm 10 & 4. \pm 3 \\
5. \pm 8 & 5. \pm 50 \\
6. \pm 3 & 6. \pm 20 \\
7. \pm 2 & 7. \pm 9 \\
8. \pm 7 & 8. \pm 8 \\
9. \pm 60 & 9. 1, -7 \\
10. \pm 110 & 10. -3, -15 \\
11. \pm 120 & 11. 13, -5 \\
12. \pm 100 & 12. 23, 1 \\
\end{array}
}