Question

Write a number of quadratic equations with real roots which do not change by squaring their roots

Answer 1

Let the equation is,

ax2+bx+c=0ax2+bx+c=0

or a2x2+abx+ac=0a2x2+abx+ac=0

And let the root are αα and ββ

We know the equation whose roots are α2α2and β2β2 is

a(x−−√)2+bx−−√+c=0a(x)2+bx+c=0

ax+bx−−√+c=0ax+bx+c=0

bx−−√=−(ax+c)bx=−(ax+c)

square both sides,

b2x=a2x2+2acx+c2b2x=a2x2+2acx+c2

a2x2+(2ac−b2)x+c2a2x2+(2ac−b2)x+c2

Since the leading coefficients are same that means,

2ac−b2=ab2ac−b2=ab

c2=acc2=ac

We get c=0c=0 or c=ac=a and putting in first eqn

case 1: c=0c=0

−b2=ab−b2=ab

So, b=0b=0 or b=−ab=−a

case 2: c=ac=a

2a2−ab−b2=02a2−ab−b2=0

2a2−2ab+ab−b2=02a2−2ab+ab−b2=0

2a(a−b)+b(a−b)=02a(a−b)+b(a−b)=0

(a−b)(2a+b)=0(a−b)(2a+b)=0

b=ab=a or b=−2ab=−2a

The equation can be,

x2=0x2=0

x2−x=0x2−x=0

x2+x+1=0x2+x+1=0

x2−2x+1=0x2−2x+1=0

So the answer turns out to be,

4