Solve the equation by using the square root property.
left parenthesis 5 p plus 3 right parenthesis squared equals 8
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Part 1
The solution set is StartSet 1 EndSet
  
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​(Simplify your answer. Type an exact​ answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

To solve the equation using the square root property, let's start with the given equation:

\[
(5p + 3)^2 = 8
\]

### Step-by-step Solution:

1. **Apply the square root property:**

   To eliminate the square on the left side, we take the square root of both sides:

   \[
   \sqrt{(5p + 3)^2} = \pm \sqrt{8}
   \]

   This simplifies to:

   \[
   5p + 3 = \pm \sqrt{8}
   \]

2. **Simplify the square root of 8:**

   \[
   \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
   \]

   So, the equation becomes:

   \[
   5p + 3 = \pm 2\sqrt{2}
   \]

3. **Isolate \( p \):**

   Subtract 3 from both sides:

   \[
   5p = -3 \pm 2\sqrt{2}
   \]

   Then, divide by 5:

   \[
   p = \frac{-3 \pm 2\sqrt{2}}{5}
   \]

### Conclusion:

The solution set is:

\[
\left\{ \frac{-3 + 2\sqrt{2}}{5}, \frac{-3 - 2\sqrt{2}}{5} \right\}
\]

These are the two exact solutions for the equation.

---

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How can you apply the square root property to solve other types of quadratic equations?
2. What are the key differences between solving quadratic equations by factoring and by using the square root property?
3. How do you simplify expressions involving square roots and radicals?
4. What are complex numbers, and how do they appear in quadratic equations?
5. Can we solve this problem using the quadratic formula? What would be the steps?

**Tip:** Always remember to consider both the positive and negative roots when using the square root property to solve equations.

Solve the equation by using the square root property: (5p + 3)^2 = 8. Simplify your answer. Type an exact answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.

Learn how to solve the quadratic equation (5p + 3)^2 = 8 using the square root property. This solution simplifies the expression to provide the exact answers using radicals. Perfect for high school students, this problem covers key algebraic concepts such as radicals and square roots.

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