Question

How do you solve `3x^2 - 12x + 2 = 0`?

Answer 1

`x = 2+-sqrt(30)/3`

Explanation:

The equation:

`3x^2-12x+2=0`

is in the standard form:

`ax^2+bx+c=0`

with `a=3`, `b=-12` and `c=2`

The discriminant `Delta` of a quadratic is given by the formula:

`Delta = b^2-4ac = (color(blue)(-12))^2-4(color(blue)(3))(color(blue)(2)) = 144-24 = 120 = 2^2*30`

Since `Delta > 0`, the given quadratic equation has two 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(120))/6`

`color(white)(x) = (12+-sqrt(2^2*30))/6`

`color(white)(x) = (12+-2sqrt(30))/6`

`color(white)(x) = 2+-sqrt(30)/3`

That is:

`x = 2+sqrt(30)/3" "` or `" "x = 2-sqrt(30)/3`