need help finding all values of j in quadratic 7x^2+4x+j=0. How do I know if <,> or =

Question

with parameter

Answer 1

7x²+4x+j = 0

 

b² - 4ac = discriminant =  4² - 4(7)j = 16 - 28j

 

16-28j = 0 when j = 4/7

 

If j < 4/7, then the equation 7x² +4x + j = 0 has two real roots.

 

If j > 4/7, then the equation 7x²+4x + j = 0 has two imaginary roots.

 

If j = 4/7, then the equation 7x² + 4x + j = 0 has only one root (of multiplicity two), which is real.

                                                                                               

Answer 2

You need to specify what condition j must satisfy.  For example, if you want j such that the roots are real, then the discriminant D must be ≥ 0.

 

D = b² - 4ac = 4² - 4(7)(j) ≥ 0

 

16 ≥ 28j

 

j ≤ 16/28 = 4/7

 

j ≤ 4/7

 

Perhaps you want j such that the two roots are equal.  Then set D = 0 and solve for j.  Perhaps you want both roots to be complex.  Then set D < 0 and solve for j.  

 

What condition do you want j to meet?