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: $x = \frac{3 + 3\sqrt{2}}{2} \quad \text{or} \quad x = \frac{3 - 3\sqrt{2}}{2}$

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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.