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

1 Answer
George C.
Mar 13, 2017

This quadratic has two distinct irrational real roots.

Explanation:

Given:

x^2=4x-1

Subtract 4x-4 from both sides to get:

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 4x-1 from both sides to get:

x^2-4x+1 = 0

This is in standard form:

ax^2+bx+c = 0

with a=1, b=-4 and c=1.

This has discriminant Delta, given by the formula:

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

Since Delta > 0 we can deduce that our quadratic has two distinct real roots.

Note also that Delta is not a perfect square, so those roots are irrational.

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