https://file.aiquickdraw.com/mathbot/file/1725370145730qhm350t2.jpg

The image contains a list of quadratic equations that need to be solved by extracting the roots. Let's go through each problem one by one.

### 1. \( x^2 = 75 \)
To solve for \( x \), take the square root of both sides:
\[
x = \pm \sqrt{75}
\]
Simplifying \( \sqrt{75} \):
\[
\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
\]
So, the solutions are:
\[
x = \pm 5\sqrt{3}
\]

### 2. \( x^2 = 80 \)
Take the square root of both sides:
\[
x = \pm \sqrt{80}
\]
Simplifying \( \sqrt{80} \):
\[
\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}
\]
So, the solutions are:
\[
x = \pm 4\sqrt{5}
\]

### 3. \( 2(x - 5)^2 = 32 \)
First, divide both sides by 2:
\[
(x - 5)^2 = 16
\]
Take the square root of both sides:
\[
x - 5 = \pm 4
\]
Solve for \( x \):
\[
x = 5 \pm 4
\]
The solutions are:
\[
x = 9 \quad \text{or} \quad x = 1
\]

### 4. \( 3(4x - 1)^2 - 1 = 11 \)
First, add 1 to both sides:
\[
3(4x - 1)^2 = 12
\]
Then, divide both sides by 3:
\[
(4x - 1)^2 = 4
\]
Take the square root of both sides:
\[
4x - 1 = \pm 2
\]
Solve for \( x \):
1. If \( 4x - 1 = 2 \):
   \[
   4x = 3 \quad \Rightarrow \quad x = \frac{3}{4}
   \]
2. If \( 4x - 1 = -2 \):
   \[
   4x = -1 \quad \Rightarrow \quad x = -\frac{1}{4}
   \]
The solutions are:
\[
x = \frac{3}{4} \quad \text{or} \quad x = -\frac{1}{4}
\]

### 5. \( (2x - 3)^2 = 18 \)
Take the square root of both sides:
\[
2x - 3 = \pm \sqrt{18}
\]
Simplify \( \sqrt{18} \):
\[
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\]
Solve for \( x \):
\[
2x = 3 \pm 3\sqrt{2}
\]
\[
x = \frac{3 \pm 3\sqrt{2}}{2}
\]
The solutions are:
\[
x = \frac{3 + 3\sqrt{2}}{2} \quad \text{or} \quad x = \frac{3 - 3\sqrt{2}}{2}
\]

---

Would you like any further details or have any questions?

### Relative Questions:
1. How can you verify the solutions to these quadratic equations?
2. What are the real-life applications of solving quadratic equations?
3. How do you solve a quadratic equation when it's not easily factorable?
4. What is the significance of the discriminant in quadratic equations?
5. How can you derive the quadratic formula from a standard quadratic equation?

**Tip:** When solving quadratic equations, always consider checking the solutions by substituting them back into the original equation to ensure they satisfy it.

Solve the quadratic equations by extracting the roots: 1. x^2 = 75 2. x^2 = 80 3. 2(x - 5)^2 = 32 4. 3(4x - 1)^2 - 1 = 11 5. (2x - 3)^2 = 18

Learn how to solve quadratic equations by extracting the roots. This tutorial covers key algebraic concepts and techniques, including the square root theorem, and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation