91Ó°ÊÓ

The quadratic formula is a reliable method to solve any quadratic equation. It's given by the formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides the roots of a quadratic equation \( ax^2 + bx + c = 0 \).
Here's how to use it:

• Identify the coefficients \(a\), \(b\), and \(c\) from the standard form.
• Substitute these coefficients into the quadratic formula.
• Calculate the discriminant \( \Delta \) (the part under the square root).
• Use the discriminant to find the roots.

In our example, we have \( a = 1 \), \( b = -6 \), and \( c = 11 \). Substituting these into the quadratic formula gives us the roots

\[ x = \frac{6 \pm \sqrt{-8}}{2} \]
These roots are simplified to \[ x = 3 \pm i\sqrt{2} \].

Quadratic equations can sometimes have complex solutions, especially when the discriminant is negative.
Complex numbers have a real part and an imaginary part and are written in the form \( a + bi \), where \( i \) is the imaginary unit. It’s defined by the property that \( i^2 = -1 \).
In our example, the discriminant is \( \Delta = -8 \), giving us complex solutions. When plugging the discriminant into the quadratic formula, we get:

\[ x = 3 \pm i\sqrt{2} \]
Here, \( 3 \) is the real part and \( \pm \sqrt{2}i \) is the imaginary part.
These solutions mean that the parabola described by the quadratic equation does not intersect the x-axis and lies entirely above or below it, depending on the sign of the leading coefficient.

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[ \Delta = b^2 - 4ac \]
The value of the discriminant determines the nature of the roots of the quadratic equation:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has one real root (a repeated root).
• If \( \Delta < 0 \), the equation has two complex conjugate roots.

In our problem, the discriminant \( \Delta \) is calculated as:

\[ \Delta = (-6)^2 - 4(1)(11) = 36 - 44 = -8 \]
Since \( \Delta = -8 \), it's negative, indicating the quadratic equation has two complex solutions.