solve using the quadratic formula. $$ 2p^2+8p+5=0 $$

To solve the quadratic equation $ 2p^2 + 8p + 5 = 0 $ using the quadratic formula, we start by identifying the coefficients $ a $, $ b $, and $ c $ in the standard form $ ax^2 + bx + c = 0 $:

\[
a = 2, \quad b = 8, \quad c = 5
\]

The quadratic formula is given by:

\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step 1: Substitute the values of $ a $, $ b $, and $ c $ into the formula
\[
p = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}
\]

### Step 2: Simplify inside the square root
\[
p = \frac{-8 \pm \sqrt{64 - 40}}{4}
\]

\[
p = \frac{-8 \pm \sqrt{24}}{4}
\]

### Step 3: Simplify the square root of 24
Since $ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} $, we have:
\[
p = \frac{-8 \pm 2\sqrt{6}}{4}
\]

### Step 4: Simplify the fraction
Divide each term by 2:
\[
p = \frac{-4 \pm \sqrt{6}}{2}
\]

### Final Answer
The solutions are:
\[
p = \frac{-4 + \sqrt{6}}{2} \quad \text{and} \quad p = \frac{-4 - \sqrt{6}}{2}
\]

Would you like more details or have any questions?

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### Related Questions
1. How does the quadratic formula apply to equations without a real solution?
2. What other methods can solve quadratic equations besides the quadratic formula?
3. Can all quadratic equations be factored, and how does factoring relate to the quadratic formula?
4. What is the importance of the discriminant $ b^2 - 4ac $ in the quadratic formula?
5. How do you determine if a quadratic equation has one, two, or no real solutions?

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**Tip:** Simplifying the square root as much as possible is essential for obtaining the most reduced form of your answers.

Learn how to solve the quadratic equation 2p^2 + 8p + 5 = 0 using the quadratic formula. This step-by-step solution walks through identifying coefficients, applying the quadratic formula, and simplifying the result, making it ideal for students in Grades 9-11.

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