91Ó°ÊÓ

The imaginary unit is a key concept in complex numbers. We denote it by the symbol \( i \). The defining property of the imaginary unit is that \( i^{2} = -1 \). This is a very useful idea for working with complex numbers and equations that involve negative square roots. By expressing the square root of a negative number in terms of \( i \), we transform an otherwise unsolvable equation into something manageable.
For example, \( \sqrt{-1} = i \), and \( \sqrt{-4} = 2i \). By getting familiar with this concept, tackling equations involving negative square roots becomes much clearer.

When solving quadratic equations, taking the square root is a common step. It helps us find the values that satisfy the equation. The square root of a number \( x \) is a value that, when multiplied by itself, gives \( x \). For example, the square root of 4 is 2, because \( 2 \times 2 = 4 \).
For negative numbers, we use the imaginary unit we discussed earlier. For example, \( \sqrt{-1} = i \). This is crucial in solving quadratic equations that do not have real solutions. When you practice, remember these points, and taking square roots will become second nature.

Quadratic equations often appear in the form \( ax^{2} + bx + c = 0 \). Solving them can sometimes involve complex numbers, especially when the equation has no real roots. The equation in this exercise was simpler, without the \( bx \) term. Here is a step-by-step approach to solving quadratic equations:
1. **Isolate the variable term**: Rearrange the equation to get the quadratic term on one side.
2. **Recognize the need for the imaginary unit**: If you end up with a negative number under the square root, use \( i \).
3. **Take the square root of both sides**: This will give you the values of the variable that satisfy the equation.

In our specific example, isolating \( a^{2} \) gave \( a^{2} = -1 \). Recognizing the imaginary unit, \( i \), helped rewrite this as \( a^{2} = i^{2} \). Finally, taking the square root yielded the solutions \( a = \pm i \). Understanding each step makes solving these equations straightforward.