How would i solve 5p²+10p+30=5 with completing the square

To solve the quadratic equation \( 5p^2 + 10p + 30 = 5 \) by completing the square, follow these steps:

### Step 1: Simplify the equation
First, subtract 5 from both sides to set the equation to 0:
\[
5p^2 + 10p + 30 - 5 = 0
\]
This simplifies to:
\[
5p^2 + 10p + 25 = 0
\]

### Step 2: Divide by the coefficient of \( p^2 \)
To make the coefficient of \( p^2 \) equal to 1, divide the entire equation by 5:
\[
p^2 + 2p + 5 = 0
\]

### Step 3: Move the constant to the other side
Now move the constant (5) to the right-hand side:
\[
p^2 + 2p = -5
\]

### Step 4: Complete the square
To complete the square, take half of the coefficient of \( p \) (which is 2), divide it by 2, and square it:
\[
\left(\frac{2}{2}\right)^2 = 1
\]
Now, add this value to both sides of the equation:
\[
p^2 + 2p + 1 = -5 + 1
\]
This simplifies to:
\[
(p + 1)^2 = -4
\]

### Step 5: Solve for \( p \)
Take the square root of both sides:
\[
p + 1 = \pm \sqrt{-4}
\]
\[
p + 1 = \pm 2i
\]
(where \( i \) is the imaginary unit).

Finally, solve for \( p \) by subtracting 1 from both sides:
\[
p = -1 \pm 2i
\]

### Final answer:
\[
p = -1 + 2i \quad \text{or} \quad p = -1 - 2i
\]

Would you like more details on any part of this process or have any questions?

Here are some related questions for further exploration:
1. How do you solve quadratic equations using the quadratic formula?
2. Can completing the square always be used to solve quadratics with real coefficients?
3. How can you derive the quadratic formula by completing the square?
4. What are the differences between completing the square and factoring in solving quadratics?
5. How do imaginary numbers affect the solutions of quadratic equations?

**Tip**: Completing the square is especially useful for converting quadratics into vertex form, which helps in graphing and finding the maximum or minimum values.

How would I solve 5p²+10p+30=5 with completing the square?

Learn how to solve the quadratic equation 5p² + 10p + 30 = 5 using the completing the square method. This step-by-step guide explores key algebraic concepts, including quadratic equations, complex numbers, and the square root of negative numbers. Suitable for students in Grades 9-12.

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