# How do you describe the nature of the roots of the equation #x^2=4x-1#?

##### 1 Answer

Mar 13, 2017

This quadratic has two distinct irrational real roots.

#### Explanation:

Given:

#x^2=4x-1#

Subtract

#x^2-4x+4=3#

That is:

#(x-2)^2 = 3#

Hence:

#x-2 = +-sqrt(3)#

So:

#x = 2+-sqrt(3)#

So this quadratic equation has two distinct irrational real roots.

Note that instead of the full derivation of the roots, we could have examined the discriminant...

Given:

#x^2=4x-1#

Subtract

#x^2-4x+1 = 0#

This is in standard form:

#ax^2+bx+c = 0#

with

This has discriminant

#Delta = b^2-4ac = (-4)^2-4(1)(1) = 16-4 = 12 = 2^2*3#

Since

Note also that