it is given that (k^2+1)x^2=3x-4 is an equation in x. Prove that the equation has no real roots for all real numbers k.

Question

it is given that (k^2+1)x^2=3x-4 is an equation in x. Prove that the equation has no real roots for all real numbers k. How do I do this I've been at it for an hour now lol what does the end have to be for me to prove it????

Answer 1

Let a= k^2+1 > 1

a is larger than +1 for any k

ax^2 = 3x-4

ax^2 -3x + 4= 0

discriminate = b^2 -4ac = 9-4a(4) = 9-16a <0 since a >1

the discriminate is always negative which means there are no real roots