Sure! Let’s solve each problem using the
square root property, which states:
> If \( x^2 = a \), then \( x = \pm\sqrt{a} \).
This applies to any expression squared equal to a number — we take the square root of both sides and include both the positive and negative roots.
---
Problem 1: \( x^2 = 27 \)
Step 1: Apply the square root property.
\[
x = \pm\sqrt{27}
\]
Step 2: Simplify the square root.
\[
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}
\]
✔ Final Answer:
\[
\boxed{x = \pm 3\sqrt{3}}
\]
---
Problem 2: \( (x + 3)^2 = 64 \)
Step 1: Take the square root of both sides.
\[
x + 3 = \pm\sqrt{64}
\]
\[
x + 3 = \pm 8
\]
Step 2: Solve for \( x \) by subtracting 3 from both sides.
- Case 1: \( x + 3 = 8 \Rightarrow x = 5 \)
- Case 2: \( x + 3 = -8 \Rightarrow x = -11 \)
✔ Final Answer:
\[
\boxed{x = 5 \quad \text{or} \quad x = -11}
\]
---
Problem 3: \( (4x + 3)^2 = 24 \)
Step 1: Take the square root of both sides.
\[
4x + 3 = \pm\sqrt{24}
\]
Step 2: Simplify \( \sqrt{24} \).
\[
\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}
\]
So,
\[
4x + 3 = \pm 2\sqrt{6}
\]
Step 3: Subtract 3 from both sides.
\[
4x = -3 \pm 2\sqrt{6}
\]
Step 4: Divide both sides by 4.
\[
x = \frac{-3 \pm 2\sqrt{6}}{4}
\]
✔ Final Answer:
\[
\boxed{x = \frac{-3 \pm 2\sqrt{6}}{4}}
\]
---
Problem 4: \( (x - 5)^2 + 36 = 0 \)
Step 1: Isolate the squared term.
Subtract 36 from both sides:
\[
(x - 5)^2 = -36
\]
Step 2: Take the square root of both sides.
\[
x - 5 = \pm\sqrt{-36}
\]
Step 3: Simplify the square root of a negative number using imaginary numbers.
\[
\sqrt{-36} = \sqrt{-1 \cdot 36} = i\sqrt{36} = 6i
\]
So,
\[
x - 5 = \pm 6i
\]
Step 4: Add 5 to both sides.
\[
x = 5 \pm 6i
\]
✔ Final Answer:
\[
\boxed{x = 5 \pm 6i}
\]
---
##
✔ Summary of All Answers:
1. \( x = \pm 3\sqrt{3} \)
2. \( x = 5 \) or \( x = -11 \)
3. \( x = \dfrac{-3 \pm 2\sqrt{6}}{4} \)
4. \( x = 5 \pm 6i \)
Let me know if you’d like to see these graphed or explained differently!
Parent Tip: Review the logic above to help your child master the concept of college math worksheet.
