Question 1: Solve \( x^2 = \frac{25}{64} \)
To find \( x \), we take the square root of both sides:
\[ x = \pm \sqrt{\frac{25}{64}} \]
Calculating the square root:
\[ \sqrt{\frac{25}{64}} = \frac{\sqrt{25}}{\sqrt{64}} = \frac{5}{8} \]
Thus, the two solutions are:
\[ x = \frac{5}{8} \quad \text{and} \quad x = -\frac{5}{8} \]
Listing the answers from smallest to largest:
- The first solution is \( x = -\frac{5}{8} \)
- The second solution is \( x = \frac{5}{8} \)
Question 2: Solve the equation \( \frac{(x + 27)^2}{-6} = -3 \)
First, we clear the fraction by multiplying both sides by -6:
\[ (x + 27)^2 = -3 \times -6 \] \[ (x + 27)^2 = 18 \]
Now, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{18} \]
Calculating \( \sqrt{18} \):
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
Therefore, we have:
\[ x + 27 = 3\sqrt{2} \quad \text{or} \quad x + 27 = -3\sqrt{2} \]
Now, we solve for \( x \):
- From \( x + 27 = 3\sqrt{2} \):
\[ x = 3\sqrt{2} - 27 \]
- From \( x + 27 = -3\sqrt{2} \):
\[ x = -3\sqrt{2} - 27 \]
Next, we approximate \( \sqrt{2} \), which is approximately 1.414. Thus:
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\( 3\sqrt{2} \approx 3 \times 1.414 \approx 4.242 \) So, \[ x \approx 4.242 - 27 \approx -22.758 \]
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\( -3\sqrt{2} \approx -4.242 \) So, \[ x \approx -4.242 - 27 \approx -31.242 \]
Rounding to the nearest hundredth:
- \( x \approx -22.76 \)
- \( x \approx -31.24 \)
Listing the answers from smallest to largest:
- The first solution is \( x \approx -31.24 \)
- The second solution is \( x \approx -22.76 \)
Final Answers:
-
Question 1:
- First solution: \( x = -\frac{5}{8} \)
- Second solution: \( x = \frac{5}{8} \)
-
Question 2:
- First solution: \( x \approx -31.24 \)
- Second solution: \( x \approx -22.76 \)
1 answer