91Ó°ÊÓ

When we encounter the square root of a negative number, like in the expression \(\textstyle \sqrt{-11}\), it can seem tricky at first glance.
However, mathematicians have developed a way to handle these situations by introducing imaginary numbers.
The key idea is that the square root of a negative number involves the imaginary unit, denoted as 'i'.
By definition, \(\textstyle i\) is equal to \(\textstyle \sqrt{-1}\).
Therefore, the square root of \(\textstyle -11\) can be expressed as:
\(\textstyle \sqrt{-11} = \sqrt{11} \, \cdot \, i = i \sqrt{11}\).

Imaginary numbers extend our number system beyond the real numbers, allowing us to handle roots of negative numbers with ease.

Complex numbers contain both a real part and an imaginary part.
They are typically written in the form \(\textstyle a + bi\).
Here, \(\textstyle a\) represents the real part, and \(\textstyle bi\) represents the imaginary part.
In the given problem, the expression \(\textstyle -3 + i \sqrt{11}\) looks complicated at first, but we can break it down:

• The term \(\textstyle -3\) is the real part, so \(\textstyle a = -3\).
• The term \(\textstyle i \sqrt{11}\) includes the imaginary unit \(\textstyle i\) and the real number \(\textstyle \sqrt{11}\), so the imaginary part is \(\textstyle b = \sqrt{11}\).

This allows us to clearly identify and work with each part of a complex number.

When dealing with algebraic expressions, it's important to simplify them step by step.
Let’s revisit the problem: \(\textstyle -3 + \sqrt{3^2 - 4(1)(5)}\).
Breaking it down, we first need to simplify inside the square root: \(\textstyle 3^2 - 4(1)(5)\).
This simplifies to \(\textstyle 9 - 20 = -11\).
The expression now looks like \(\textstyle -3 + \sqrt{-11}\).
Next, we handle the square root of \(\textstyle -11\) as discussed above, converting it to the imaginary number \(\textstyle i \sqrt{11}\).
Finally, combining these simplifications, we write the expression as: \(\textstyle -3 + i \sqrt{11}\).
By following these steps: simplify under the square root, recognize imaginary parts, and rewrite it in simpler forms like \(\textstyle a + bi\), we make complex expressions more manageable.