9th Grade Math Review Worksheet featuring algebra and geometry problems.

Let's solve each problem step by step.

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Problem 1: Simplify \( \sqrt{512x^2} \)

#### Solution:
1. Factorize 512 into its prime factors:
\[
512 = 2^9
\]
So, \( \sqrt{512} = \sqrt{2^9} \).

2. Use the property of square roots: \( \sqrt{a^b} = a^{b/2} \):
\[
\sqrt{2^9} = 2^{9/2} = 2^4 \cdot 2^{1/2} = 16\sqrt{2}
\]

3. Now consider the \( x^2 \) term:
\[
\sqrt{x^2} = |x|
\]
(The square root of \( x^2 \) is the absolute value of \( x \)).

4. Combine the results:
\[
\sqrt{512x^2} = 16\sqrt{2} \cdot |x| = 16|x|\sqrt{2}
\]

#### Final Answer:
\[
\boxed{16|x|\sqrt{2}}
\]

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Problem 2: Solve \( x - 18 = -5 \)

#### Solution:
1. Add 18 to both sides of the equation to isolate \( x \):
\[
x - 18 + 18 = -5 + 18
\]
\[
x = 13
\]

#### Final Answer:
\[
\boxed{13}
\]

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Problem 3: Solve \( \sqrt{x - 8} = \sqrt{-x} + 2 \)

#### Solution:
1. Square both sides to eliminate the square roots:
\[
(\sqrt{x - 8})^2 = (\sqrt{-x} + 2)^2
\]
\[
x - 8 = (\sqrt{-x})^2 + 2 \cdot 2 \cdot \sqrt{-x} + 2^2
\]
\[
x - 8 = -x + 4\sqrt{-x} + 4
\]

2. Simplify and rearrange terms:
\[
x - 8 = -x + 4\sqrt{-x} + 4
\]
\[
x + x - 8 - 4 = 4\sqrt{-x}
\]
\[
2x - 12 = 4\sqrt{-x}
\]

3. Divide both sides by 4:
\[
\frac{2x - 12}{4} = \sqrt{-x}
\]
\[
\frac{x - 6}{2} = \sqrt{-x}
\]

4. Square both sides again to eliminate the square root:
\[
\left( \frac{x - 6}{2} \right)^2 = (\sqrt{-x})^2
\]
\[
\frac{(x - 6)^2}{4} = -x
\]

5. Multiply through by 4 to clear the fraction:
\[
(x - 6)^2 = -4x
\]

6. Expand \( (x - 6)^2 \):
\[
x^2 - 12x + 36 = -4x
\]

7. Rearrange into standard quadratic form:
\[
x^2 - 12x + 4x + 36 = 0
\]
\[
x^2 - 8x + 36 = 0
\]

8. Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -8 \), and \( c = 36 \):
\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}
\]
\[
x = \frac{8 \pm \sqrt{64 - 144}}{2}
\]
\[
x = \frac{8 \pm \sqrt{-80}}{2}
\]
\[
x = \frac{8 \pm 4i\sqrt{5}}{2}
\]
\[
x = 4 \pm 2i\sqrt{5}
\]

Since the solutions are complex numbers, there are no real solutions to this equation.

#### Final Answer:
\[
\boxed{\text{No real solutions}}
\]

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Problem 4: Find the value of \( x \). The distance between \( (4, x) \) and \( (5, 8) \) is \( \sqrt{15} \)

#### Solution:
1. Use the distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Here, \( (x_1, y_1) = (4, x) \) and \( (x_2, y_2) = (5, 8) \), and the distance \( d = \sqrt{15} \).

2. Substitute the coordinates into the distance formula:
\[
\sqrt{15} = \sqrt{(5 - 4)^2 + (8 - x)^2}
\]
\[
\sqrt{15} = \sqrt{1^2 + (8 - x)^2}
\]
\[
\sqrt{15} = \sqrt{1 + (8 - x)^2}
\]

3. Square both sides to eliminate the square root:
\[
15 = 1 + (8 - x)^2
\]

4. Subtract 1 from both sides:
\[
15 - 1 = (8 - x)^2
\]
\[
14 = (8 - x)^2
\]

5. Take the square root of both sides:
\[
\sqrt{14} = 8 - x \quad \text{or} \quad -\sqrt{14} = 8 - x
\]

6. Solve for \( x \) in each case:
- For \( \sqrt{14} = 8 - x \):
\[
x = 8 - \sqrt{14}
\]
- For \( -\sqrt{14} = 8 - x \):
\[
x = 8 + \sqrt{14}
\]

#### Final Answer:
\[
\boxed{8 \pm \sqrt{14}}
\]

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Problem 5: Simplify the following expression: \( x^2 + 18x + 72 \)

#### Solution:
1. Check if the quadratic expression can be factored. Look for two numbers that multiply to 72 and add to 18.
- The numbers are 6 and 12 because \( 6 \cdot 12 = 72 \) and \( 6 + 12 = 18 \).

2. Rewrite the expression using these numbers:
\[
x^2 + 18x + 72 = (x + 6)(x + 12)
\]

#### Final Answer:
\[
\boxed{(x + 6)(x + 12)}
\]

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Summary of Answers:

1. \( \boxed{16|x|\sqrt{2}} \)
2. \( \boxed{13} \)
3. \( \boxed{\text{No real solutions}} \)
4. \( \boxed{8 \pm \sqrt{14}} \)
5. \( \boxed{(x + 6)(x + 12)} \)