Question

if roots of the quadratic equation x^2×2px+mn=0 are real and equal, show that the roots of the quadratic equation x^2-2 (m+n)x+(m^2+n^2+2p^2)=0 are also equal

Answer 1

Hey mate..
========

The solution is in the attachment .

Hope it helps !!

Answer 2

Answer:

As shown below

Step-by-step explanation:  

Given that  $x^{2} +2px +mn =0$ has real and equal roots

here we have to prove the roots of  the equation $x^{2} -2(m+n)x +(m^{2} + n^{2} + 2p^{2}) =0$  are also equal and real

⇒ compare the equation $x^{2} + 2px +mn =0$ with $ax^{2} + bx +c =0$

⇒ $a =$ 1, $b =2p$ and $c =mn$

from data the above equation has equal and real roots

⇒ $b^{2} -4ac =0$  

⇒ $(2p)^{2} - 4(1) (mn) =0$

⇒ $4p^{2} -4mn =0$

⇒ $p^{2} -mn =0$  _ (1)

now take given 2nd equation

$x^{2} - 2(m+n)x + m ^{2} +n ^{2} + 2p ^{2} =0$  compare with  $ax^{2} + bx+c =0$

⇒ $a =1$ , $b = -2(m+n)$ and $c = m^{2} + n ^{2} + 2p ^{2}$  

if the equation have real and equal roots

⇒ $b ^{2}-4ac =0$  

⇒ $[-2(m+n)]^{2}$ - $4(1) ( m ^{2} +n ^{2} +2p ^{2} )$

⇒ $4 (m^{2} + n ^{2}+2mn) - 4 ( m^{2}+ n^{2} + 2p ^{2} )$

⇒ $4m^{2} + 4n^{2} + 8mn - 4 m^{2} -4n^{2} -8p ^{2}$  

⇒ $8mn - 8p^{2}$  

⇒ $8mn - 8mn =0$  [ from (1) $p^{2} = mn$ ]

∴  the given equation will have equal and real roots