Question

How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given `x^2-x+6=0`?

Answer 1

`x = 1/2+-sqrt(23)i`

Explanation:

The discriminant `Delta` of a quadratic equation with rational coefficients gives us the following information:

• If `Delta > 0` then there are two, distinct, real roots. If `Delta` is a perfect square, then they are both rational.

• If `Delta = 0` then there is one, repeated, rational, real root.

• If `Delta < 0` then there are two, distinct, non-Real, Complex roots, which form a complex conjugate pair.

The quadratic equation:

`x^2-x+6=0`

is in the form:

`ax^2+bx+c=0`

with `a = 1`, `b=-1` and `c=6`

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = (color(blue)(-1))^2-4(color(blue)(1))(color(blue)(6)) = 1-24 = -23`

Since `Delta < 0` this quadratic has no Real roots. It has a complex conjugate pair of non-Real, Complex roots.

We can find the roots using the quadratic formula:

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`color(white)(x) = (-b+-sqrt(Delta))/(2a)`

`color(white)(x) = (1+-sqrt(-23))/(2*1)`

`color(white)(x) = (1+-sqrt(23)i)/2`

`color(white)(x) = 1/2+-sqrt(23)i`