Question

How many quadrartic equations are possilbe which remain unchanged when the roots are squared

Answer 1

4

METHOD 1
Since we square the roots and equation does not change, that means squaring the root does not change it,

Therefore, the possibilities are

(0,0)
(1,0)
(1,1)
(w,w2) —————— Since we know (w2)2=w
Where w=1–√3 and w≠1

METHOD 2
Let the equation is,

ax2+bx+c=0

or a2x2+abx+ac=0

And let the root are α and β

We know the equation whose roots are α2 and β2 is

a(x−−√)2+bx−−√+c=0

ax+bx−−√+c=0

bx−−√=−(ax+c)

square both sides,

b2x=a2x2+2acx+c2

a2x2+(2ac−b2)x+c2

Since the leading coefficients are same that means,

2ac−b2=ab
c2=ac
We get c=0 or c=a and putting in first eqn

case 1: c=0

−b2=ab

So, b=0 or b=−a

case 2: c=a

2a2−ab−b2=0

2a2−2ab+ab−b2=0

2a(a−b)+b(a−b)=0

(a−b)(2a+b)=0

b=a or b=−2a

The equation can be,

x2=0
x2−x=0
x2+x+1=0
x2−2x+1=0
So the answer turns out to be,

4