91Ó°ÊÓ

Isolating variables is a crucial step in solving equations. It involves manipulating the equation to get the variable of interest alone on one side. Let's see how this works in our example. We started with the equation in standard form:
\( 4d^2 + 81 = 0 \)
Since our goal is to solve for \(d\), we need to isolate \(d^2\). First, we subtract 81 from both sides:
\( 4d^2 = -81 \)
Notice we performed the same operation on both sides of the equation, which is essential to maintain equality. This step leaves us with only the \(d^2\) term on one side, making it easier to proceed.
The next step is to divide by the coefficient of \(d^2\), which is 4. By doing this, we simplify the equation:
\( d^2 = -\frac{81}{4} \)
Now that \(d^2\) is isolated, we can move to the next step.

Taking square roots helps us solve for a variable that’s squared. In our equation, after isolating \(d^2\), we have:
\( d^2 = -\frac{81}{4} \)
To find \(d\), we take the square root of both sides. Remember to include both the positive and negative roots since squaring either gives the original number:
\( d = \pm \sqrt{-\frac{81}{4}} \)
Using the property of square roots, \sqrt{a/b} = \sqrt{a}/\sqrt{b}, we can break down our expression:
\( d = \pm \frac{9i}{2} \)
This step is crucial to handle correctly, as it introduces the concept of complex numbers, which we’ll explore next.

Complex numbers are essential when dealing with square roots of negative numbers. They are numbers of the form \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\).
In our problem, when we take the square root of \(-\frac{81}{4}\), we get:
\( d = \pm \sqrt{-\frac{81}{4}} \)
We know that \sqrt{-1} = i\, so we can write:
\( d = \pm \frac{9i}{2} \)
Complex numbers are not more complicated, just different. They work like regular numbers but involve the imaginary unit \(i\).
Understanding how to use and handle them opens up a wider range of solutions in mathematics.