Not so Subtle Roots

Algebra Level 3

$x^2+x+1 = 0$

The roots of the equation above are denoted by $r$ and $s$ , while the roots of th equation below are denoted by $r^3$ and $s^3$ .

$ax^2+bx+c = 0$

What is the value of $- \frac b a$ ?

The answer is 2.

2 solutions

Sujoy Roy
Nov 25, 2014

Here $r$ and $s$ are complex cube roots of unity.

So, $r^3=1$ and $s^3=1$ .

Now, Sum of the roots $=-\frac{b}{a}=\boxed{2}$

@sujoy roy .. Please explain the 'COMPLEX CUBE ROOTS OF UNITY' term. Never heard of it :(

Hrishik Mukherjee - 6 years, 2 months ago

It's the roots of 1 in cube root, which includes complex numbers by the way.

(Sorry, my teacher also didn't explain much about this.)

Joeie Christian Santana - 6 years, 2 months ago

Paul Ryan Longhas
Nov 25, 2014

x^{2} + x +1= 0 ---> r +s = -1 and rs = 1 and ax^{2} + bx + c = 0 ---> -b/a = r^3 + s^3 . So, r^3 + s^3 = (r+s)^3 - 3rs(r+s) ---> (-1)^3 -3(1)(-1)= -1+3=2 answer

Not so Subtle Roots

The answer is 2.

2 solutions

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