Question

How do you find the discriminant of `x^2-12x+4=0`?

Answer 1

`Delta = b^2-4ac`

Explanation:

`x^2-12x+4 = 0`

is in the form:

`ax^2+bx+c = 0`

with `a=1`, `b=-12` and `c=4`

It has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = (-12)^2-4(1)(4) = 144-16 = 128 = 2*8^2`

Since `Delta > 0` this quadratic equation has two distinct Real roots, but since `Delta` is not a perfect square, those roots are irrational.

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) = (12+-sqrt(2*8^2))/2`

`color(white)(x) = (12+-8sqrt(2))/2`

`color(white)(x) = 6+-4sqrt(2)`