https://file.aiquickdraw.com/mathbot/file/1729337210681hz8kcchs.jpg

The equation provided in the image is:

\[
P^2 - 2\sqrt{30} - 1 = 0
\]

To solve for \( P \), let's rearrange the equation and use the quadratic formula. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

Here, \( a = 1 \), \( b = 0 \), and \( c = -(2\sqrt{30} + 1) \).

### Applying the Quadratic Formula

The quadratic formula states:

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

Substitute the values \( a = 1 \), \( b = 0 \), and \( c = -(2\sqrt{30} + 1) \):

\[
P = \frac{-0 \pm \sqrt{0^2 - 4(1)(-(2\sqrt{30} + 1))}}{2(1)}
\]

This simplifies to:

\[
P = \pm \sqrt{4(2\sqrt{30} + 1)}
\]

\[
P = \pm 2\sqrt{2\sqrt{30} + 1}
\]

Thus, the solutions for \( P \) are:

\[
P = 2\sqrt{2\sqrt{30} + 1} \quad \text{and} \quad P = -2\sqrt{2\sqrt{30} + 1}
\]

Would you like a deeper explanation or have any questions?

### Related Questions:
1. How do you apply the quadratic formula to solve different types of quadratic equations?
2. What is the significance of the discriminant in the quadratic formula?
3. How can you solve quadratic equations by completing the square?
4. What are other methods to solve quadratic equations apart from the quadratic formula?
5. How do real and complex roots differ in quadratic equations?

### Tip:
Always check if a quadratic equation can be simplified or factored before using the quadratic formula; it can save time and effort.

Learn how to solve the quadratic equation P^2 - 2√30 - 1 = 0 using the quadratic formula. This problem involves key concepts in algebra and is ideal for students in Grades 10-12.

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