91Ó°ÊÓ

Solving quadratic equations often leads to solutions that involve complex numbers, especially when dealing with negative square roots.
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form:

• \(a + bi\)

where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit \(i\) is defined by the property that \(i^2 = -1\).
This means that when you take the square root of a negative number, you will get an imaginary number.
For example, \(\sqrt{-4} = 2i\) because \((2i)^2 = 4i^2 = -4\).
In our exercise, the solutions are real and do not involve imaginary numbers since we are taking the square root of a positive number.

The square root is a fundamental concept in solving quadratic equations.
It is the number that, when multiplied by itself, gives the original number.
For instance, \( \sqrt{9}=3 \) and \(-3 \) because \(3 \cdot 3 \) or \((-3) \cdot (-3) = 9 \).
When isolating the variable \( p \) in our equation \( p^{2} = 50 \), we take the square root of both sides to solve for \( p \):

• \(p = \pm \sqrt{50} = \pm 5 \sqrt{2} \)

It is crucial to consider both the positive and negative solutions because squaring either a positive or negative number will give a positive outcome.
Furthermore, simplifying \( \sqrt{50} \) into \( \sqrt{25 \cdot 2} = 5 \sqrt{2} \) makes it easier to work with and better shows the relationship of the square root in the equation.

Verification is an essential step in solving quadratic equations as it ensures that the solutions are correct.
By substituting the obtained solutions back into the original equation, we can confirm their validity.
For our solutions \( p = 5\sqrt{2} \) and \( p = -5\sqrt{2} \), the verification process would be as follows:

• Substitute \( p = 5\sqrt{2} \):
  \((5\sqrt{2})^{2} - 50 = 0 \)
  \(50 - 50 = 0\)
• Substitute \( p = -5\sqrt{2} \):
  \((-5\sqrt{2})^{2} - 50 = 0 \)
  \(50 - 50 = 0\)

Both solutions satisfy the original equation, proving that they are correct.
Always verify your solutions to avoid mistakes and to make sure that you've thoroughly understood the problem.