Alright, let's solve each of the quadratic equations by extracting square roots. Here's a detailed, step-by-step solution for each one:
### 1. [tex]\( x^2 = 18 \)[/tex]
To solve for [tex]\( x \)[/tex], we need to take the square root of both sides of the equation:
[tex]\[ x = \pm \sqrt{18} \][/tex]
The positive and negative square roots of 18 are:
[tex]\[ x = \pm 4.242640687119285 \][/tex]
So, the solutions are:
[tex]\[ x = 4.242640687119285 \quad \text{and} \quad x = -4.242640687119285 \][/tex]
### 2. [tex]\( x^2 = 32 \)[/tex]
Similarly, we take the square root of both sides:
[tex]\[ x = \pm \sqrt{32} \][/tex]
The positive and negative square roots of 32 are:
[tex]\[ x = \pm 5.656854249492381 \][/tex]
Thus, the solutions are:
[tex]\[ x = 5.656854249492381 \quad \text{and} \quad x = -5.656854249492381 \][/tex]
### 3. [tex]\( x^2 = 75 \)[/tex]
Taking the square root of both sides:
[tex]\[ x = \pm \sqrt{75} \][/tex]
The positive and negative square roots of 75 are:
[tex]\[ x = \pm 8.660254037844387 \][/tex]
Therefore, the solutions are:
[tex]\[ x = 8.660254037844387 \quad \text{and} \quad x = -8.660254037844387 \][/tex]
### 4. [tex]\( x^2 = 28 \)[/tex]
We take the square root of both sides:
[tex]\[ x = \pm \sqrt{28} \][/tex]
The positive and negative square roots of 28 are:
[tex]\[ x = \pm 5.291502622129181 \][/tex]
So, the solutions are:
[tex]\[ x = 5.291502622129181 \quad \text{and} \quad x = -5.291502622129181 \][/tex]
### 5. [tex]\( x^2 = 242 \)[/tex]
Taking the square root of both sides:
[tex]\[ x = \pm \sqrt{242} \][/tex]
The positive and negative square roots of 242 are:
[tex]\[ x = \pm 15.556349186104045 \][/tex]
Thus, the solutions are:
[tex]\[ x = 15.556349186104045 \quad \text{and} \quad x = -15.556349186104045 \][/tex]
### 6. [tex]\( x^2 = \frac{1}{16} \)[/tex]
Taking the square root of both sides:
[tex]\[ x = \pm \sqrt{\frac{1}{16}} \][/tex]
The positive and negative square roots of [tex]\(\frac{1}{16}\)[/tex] are:
[tex]\[ x = \pm 0.25 \][/tex]
Therefore, the solutions are:
[tex]\[ x = 0.25 \quad \text{and} \quad x = -0.25 \][/tex]
These steps provide you with a clear understanding of how to solve each quadratic equation by extracting square roots, leading to the calculated solutions.
