91Ó°ÊÓ

When solving quadratic equations, sometimes we encounter negative values under the square root. In real numbers, you cannot take the square root of a negative value, but this problem leads us to imaginary numbers. Imaginary numbers make it possible to take square roots of negative values.

Consider the equation we solved: \ x^2 = -2
Taking the square root of both sides gives us: \ x = \pm \sqrt{-2}
The square root of -1 is represented by the imaginary unit, denoted as \(i\). Thus, \(\sqrt{-2}\) becomes \( \sqrt{2} \cdot \sqrt{-1}\), which simplifies to \(\pm i\sqrt{2}\).

The solutions to \(5x^2 + 10 = 0\) are \(x = \pm i\sqrt{2}\). Keep in mind that

• Imaginary numbers are expressed as a real number multiplied by \(i\).
• They extend our number system to solve equations that have no real solutions.

This broadens the horizon for still tackling complex problems, but in another dimension of numbers.

The first step in solving the equation \(5x^2 + 10 = 0\) involves isolating the quadratic term.

This means we want the term containing \(x^2\) on one side and everything else on the other. Here’s how to do it:

• Start by using the subtraction to move constants from the left side.

Subtract 10 from both sides:
\[ 5x^2 + 10 - 10 = 0 - 10 \]
This simplifies to:
\[ 5x^2 = -10 \]
The quadratic term \(5x^2\) is now isolated. This isolation makes it easier to handle, especially when it comes to simpler steps like dividing by the coefficient to solve the equation.

Once the quadratic term is isolated, the next step is to divide by the coefficient of \(x^2\). The coefficient in our case is 5.

Here’s the reason why this step is crucial:

• The goal is to simplify the equation such that we have \(x^2\) alone on one side.
• Dividing by the coefficient helps maintain objective results and account for the value of \(x^2\) accurately.

Take the isolated quadratic term equation:
\[ 5x^2 = -10 \]
Divide both sides by 5:
\[ \frac{5x^2}{5} = \frac{-10}{5} \]
This simplifies to:
\[ x^2 = -2 \]
Now, \(x^2\) is cleanly isolated as -2, ready for additional solving steps like finding the square root and handling any imaginary number considerations.