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solve each quadratic equation by the square roots method. simplify all irrational solutions.
1. ( x^2 - 64 = 0 )
2. ( 2x^2 - 18 = 0 )
3. ( 5x^2 - 20 = 0 )
4. ( x^2 + 81 = 0 )
5. ( x^2 - 24 = 57 )
6. ( 2x^2 - 22 = 28 )
7. ( \frac{2}{3}x^2 = 24 )
8. ( \frac{1}{4}x^2 - 1 = 15 )
9. ( 4x^2 - 25 = 0 )
10. ( 9x^2 = 49 )
© gina wilson (all things algebra®, llc). 2012-2017

Question

Accepted Answer

### Problem 1: $ x^2 - 64 = 0 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 64 to both sides: $ x^2 = 64 $
## Step 2: Take square roots
$ x = \pm\sqrt{64} = \pm 8 $
# Answer: $ x = 8 $ or $ x = -8 $

### Problem 2: $ 2x^2 - 18 = 0 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 18: $ 2x^2 = 18 $, divide by 2: $ x^2 = 9 $
## Step 2: Take square roots
$ x = \pm\sqrt{9} = \pm 3 $
# Answer: $ x = 3 $ or $ x = -3 $

### Problem 3: $ 5x^2 - 20 = 0 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 20: $ 5x^2 = 20 $, divide by 5: $ x^2 = 4 $
## Step 2: Take square roots
$ x = \pm\sqrt{4} = \pm 2 $
# Answer: $ x = 2 $ or $ x = -2 $

### Problem 4: $ x^2 + 81 = 0 $
# Explanation:
## Step 1: Isolate $ x^2 $
Subtract 81: $ x^2 = -81 $
## Step 2: Analyze square roots
Square of real number is non - negative, so $ x = \pm\sqrt{-81}=\pm 9i $ (complex solutions)
# Answer: $ x = 9i $ or $ x = -9i $ (if complex allowed) or no real solutions

### Problem 5: $ x^2 - 24 = 57 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 24: $ x^2 = 57 + 24 = 81 $
## Step 2: Take square roots
$ x = \pm\sqrt{81}=\pm 9 $
# Answer: $ x = 9 $ or $ x = -9 $

### Problem 6: $ 2x^2 - 22 = 28 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 22: $ 2x^2 = 28 + 22 = 50 $, divide by 2: $ x^2 = 25 $
## Step 2: Take square roots
$ x = \pm\sqrt{25}=\pm 5 $
# Answer: $ x = 5 $ or $ x = -5 $

### Problem 7: $ \frac{2}{3}x^2 = 24 $
# Explanation:
## Step 1: Isolate $ x^2 $
Multiply both sides by $ \frac{3}{2} $: $ x^2 = 24\times\frac{3}{2}=36 $
## Step 2: Take square roots
$ x = \pm\sqrt{36}=\pm 6 $
# Answer: $ x = 6 $ or $ x = -6 $

### Problem 8: $ \frac{1}{4}x^2 - 1 = 15 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 1: $ \frac{1}{4}x^2 = 16 $, multiply by 4: $ x^2 = 64 $
## Step 2: Take square roots
$ x = \pm\sqrt{64}=\pm 8 $
# Answer: $ x = 8 $ or $ x = -8 $

### Problem 9: $ 4x^2 - 25 = 0 $
# Explanation:
## Step 1: Isolate $ x^2 $
Add 25: $ 4x^2 = 25 $, divide by 4: $ x^2=\frac{25}{4} $
## Step 2: Take square roots
$ x=\pm\sqrt{\frac{25}{4}}=\pm\frac{5}{2} $
# Answer: $ x=\frac{5}{2} $ or $ x = -\frac{5}{2} $

### Problem 10: $ 9x^2 = 49 $
# Explanation:
## Step 1: Isolate $ x^2 $
Divide by 9: $ x^2=\frac{49}{9} $
## Step 2: Take square roots
$ x=\pm\sqrt{\frac{49}{9}}=\pm\frac{7}{3} $
# Answer: $ x=\frac{7}{3} $ or $ x = -\frac{7}{3} $

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QUESTION IMAGE

Question

Explanation:

Problem 1: \( x^2 - 64 = 0 \)

Step 1: Isolate \( x^2 \)

Step 2: Take square roots

Step 1: Isolate \( x^2 \)

Step 2: Take square roots

Step 1: Isolate \( x^2 \)

Step 2: Take square roots

Answer:

Problem 2: \( 2x^2 - 18 = 0 \)