Condition for roots of the equation to be real.

Question

Show that for $ 3 > y_1 >0 $ the roots of the equation

$$(y_1-2)x^2-(8-2y_1)x-(8-3y_1)=0$$ are real, where $y_1$ is a constant.

Due to my difficulties in doing this I would be grateful for your help.

Answer 1

The question asks us to show if the roots of the equation (y1−2)x^2−(8−2y1)x−(8−3y1)=0 are real when 3>y1>0 which means:

 b^2-4ac>0 or b^2-4ac=0  when y1 is any number between 0 and 3

Therefore: we can replace y1 by 0, 1, 2 or 3 and it will give an answer like zero or a number which is greater than 0.

For e.g Lets take y1= 1

Replace in the equation which will give

        -x^2-6-5=0

Using b^2-4ac

  = (-6)^2-4(-1)(-5)

  = 36-20

  = 16            

Since 16>0 the roots of the equation are real.