For what values of c will the equation 9x212xc0 have complex

For what values of c will the equation 9x^2-12x+c=0 have complex roots?

Solution

The given equation is 9x² - 12x + c = 0. Its roots are [ -(-12) ± { ( -12)² - 4*9 *c} ]/ 2*9 = [ 12 ± ( 144 - 36c) ] / 18. Thus, the given equation will have complex roots if 144 -3 6c < 0 i.e. if 36c > 144 or c > 144/36 or, c > 4. In such a situation  (144 - 36 c) will be imaginary so that the roots of the given equation will be complex numbers.

Note: The roots of the qudratic equation ax + bx + c = 0 are [ - b ± ( b² - 4ac)] / 2a