Question

How do you solve `x^2 - 12x + 52 = 0`?

Answer 1

`x^2 - 12x + 52 = 0`

The equation is of the form `color(blue)(ax^2+bx+c=0` where:
`a=1, b=-12, c=52`

The Discriminant is given by:
`Delta=b^2-4*a*c`
`= (-12)^2-(4*1*52)`
`= 144 - 208`
`= -64`

If `Delta=0` then there is only one solution.
(for `Delta>0` there are two solutions,
for `Delta<0` there are no real solutions)

As `Delta = -64`, this equation has NO REAL SOLUTIONS

Note :
The solutions are normally found using the formula
`x=(-b+-sqrtDelta)/(2*a)`

As `Delta = -64`, `x = (-(-12)+-sqrt(-64))/(2*1) = (12+-sqrt(-64))/2`

Answer 2

`x^2-12x+52` is of the form `ax^2+bx+c` with `a=1`, `b=-12` and `c=52`. This has discriminant given by the formula:

`Delta = b^2-4ac = (-12)^2-(4xx1xx52)`

`=144-208=-64 < 0`

So `x^2-12x+52=0` has no real roots. It has two distinct complex roots.