91Ó°ÊÓ

To solve any quadratic equation of the form \(ax^2 + bx + c = 0\), we can use the quadratic formula. This remarkable formula provides the solutions (or roots) for the given quadratic equation.
Here's the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

• Identify the coefficients \(a\), \(b\), and \(c\). They are constants in the equation's format.
• Substitute these coefficients into the formula.
• Simplify the equation to find the values of \(x\).

Let's use this process on the equation: \(x^2 - 4x + 9 = 0\). Here, \(a = 1\), \(b = -4\), and \(c = 9\). Using the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \].
Simplifying inside the square root, we find \(16 - 36 = -20\). This tells us more about the kinds of solutions we have.

An imaginary number arises when we attempt to take the square root of a negative number. Typically, the square root of a negative number isn't a real number. Instead, it's termed imaginary.
In our formula, the \(\sqrt{-20}\) presents an issue since \(20\) is negative. By convention, we use the imaginary unit \(i\), where \(i = \sqrt{-1}\).
So \(\sqrt{-20} = \sqrt{20}i\). We can further simplify \((\sqrt{20})\) as \(2\sqrt{5}\) because \(20 = 4 \times 5\), and \(\sqrt{4} = 2\). Therefore, \[ \sqrt{-20} = 2\sqrt{5}i \].
Imaginary numbers help us handle roots of negative quantities intuitively.

Complex solutions emerge when the quadratic equation doesn't intersect the x-axis. This happens when the discriminant (\b^2 - 4ac) is negative.
Complex numbers are formed from a real part and an imaginary part. In our equation: \[ x = \frac{4 \pm 2 \sqrt{5}i}{2} \].
Simplifying, we get: \[ x = 2 \pm \sqrt{5}i \].
Here, \(x\) has both a real part \(2\) and an imaginary part \(\pm \sqrt{5}i\). Complex solutions are written as \[ a + bi \] where \(a\) (real part) is \(2\) and \(bi\) (imaginary part) is \(\sqrt{5}i\).
Complex solutions play a crucial role in various fields, including engineering and physics, by simplifying calculations involving oscillations and waves.
So, the final solutions \(x = 2 \pm \sqrt{5}i\) elucidate both the real and imaginary aspects of the complex solutions.