Question

Find the values of $a$ for which the equation $x^2+(1-2a)x^2+a^2-1=0$ (a) has no solution; (b) has two solutions; (c) has three solutions.

Answer 1

$x^2+(1-2a)x^2+a^2-1=0$

$(2-2a)x^2+a^2-1=0$

For no solution $D<0$.

So,$0-4.2(1-a)(a^2-1)<0$

$(a-1{)}^2(a+1)<0$

$aϵ\varphi$

For two solution $D>0$

$(a-1{)}^2(a+1)>0$

$aϵR-\{-1,1\}$

For 3 solutions coefficient of $x^2=0$

So,$a=1$