### The quadratic formula and the discriminant

The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation \(ax^2+bx+c=0\), when they exist, in terms of the coefficients \(a,b,c\). The solutions are

\[ x = \dfrac{-b+\sqrt{b^2-4ac}}{2a}, \qquad x = \dfrac{-b-\sqrt{b^2-4ac}}{2a}, \]provided that \(b^2-4ac \geq 0\).

The quantity \(b^2-4ac\) is called the **discriminant** of the quadratic, often denoted by \(\Delta\), and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:

- \(\Delta > 0 \) tells us the equation has two distinct real roots
- \(\Delta = 0 \) tells us the equation has one (repeated) real root
- \(\Delta < 0 \) tells us the equation has no real roots.

Screencast of interactive 2 , Interactive 2

A quadratic expression which always takes positive values is called **positive definite**, while one which always takes negative values is called **negative definite**.

Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if \(a > 0\), and negative definite if \(a < 0\).

Detailed description of diagram

#### Example

Show that the quadratic expression \(4x^2-8x+7\) always takes positive values for any value of \(x\).

#### Solution

In this case, \(a=4\), \(b=-8\) and \(c=7\). So

\[ \Delta = (-8)^2 - 4\times 4 \times 7 = -48 < 0 \]and \(a=4 > 0\). Hence the quadratic is positive definite.