91Ó°ÊÓ

The quadratic formula is a powerful tool for solving quadratic equations. These are equations in the form of \(ax^2 + bx + c = 0\). The quadratic formula can be written as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]. This formula gives you the solutions (or roots) of the quadratic equation.
To use it, you just need to identify the coefficients a, b, and c from your equation.
Then, plug these values into the formula.
Don't forget to take care of the \(\pm\) symbol, which means you will generally have two solutions: one involving addition and one involving subtraction.

Complex numbers involve both real and imaginary parts.
They are written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\).
When the discriminant in the quadratic formula (\(b^2 - 4ac\)) is negative, the solution will involve complex numbers.
For instance, if \(\sqrt{-31} = i\sqrt{31}\), then the solutions are \(d = \frac{{3 \pm i\sqrt{31}}} {10}\).
This means the equation has two complex solutions: \(d = \frac{3 + i\sqrt{31}}{10}\) and \(d = \frac{3 - i\sqrt{31}}{10}\).

The discriminant in the quadratic formula is the part under the square root symbol, \(b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (\(b^2 - 4ac > 0\)), you will have two distinct real roots.
If it's zero (\(b^2 - 4ac = 0\)), you will have exactly one real root (or a repeated root).
If it's negative (\(b^2 - 4ac < 0\)), the roots will be complex. For example, in the equation \(5d^2 - 3d + 2 = 0\), the discriminant is \(-31\).
Since this is negative, the roots are complex, \(d = \frac{3 \pm i\sqrt{31}}{10}\).
Understanding the discriminant is crucial for predicting the type of solutions without fully solving the equation.