Prove that the roots of (a-b)^2x^2+2(a+b-2c)x+1=0 , a

Question

Prove that the roots of (a-b)^2x^2+2(a+b-2c)x+1=0 , a<b, are real or imaginary according as c does or does not lie between a and b.

anonymous · Answer

for the quadratic equation ax^2+bx+c=0 to have imaginary roots, the condition which is to be satisfied is b^2-4ac<0.
use this condition to your equation and then find the condition for c.

anonymous · Answer

$$(a-b)^2x^2+2(a+b-2c)x+1=0$$
$$b^2-4ac \implies $$
$$(2(a+b-2c))^2-4(a-b)^2*1$$

anonymous · Answer

$$4(a^2+b^2+4c^2-2(ab-2ac-2bc)-a^2+2ab-b^2)$$

anonymous · Answer

$$4(4c^2-2ab+4ac+4bc+2ab)$$

anonymous · Answer

$$4(4c^2+4ac+4bc)$$
$$16(c^2+c(a+b))$$

anonymous · Answer

this is real when $$b^2-4ac>0$$ real roots
$$c^2+c(a+b) \ge0$$
$$c+a+b \ge 0$$
$$c\ge a+b$$

anonymous · Answer

i mean 
$$a+b \ge c$$

anonymous · Answer

for imaginery
$$a+b \le c$$

anonymous · Answer

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