Expert: Ahmed Salami - 11/19/2009

Question
Hello,

I was wondering is it possible for a quadratic equation to have one real root and one imagery root! And could you explain by using examples?

And

Find k such that the graph of y=9x^2+3kx+k

a. intersects the x-axis in one point only
b.intersects that x-axis in two points
c. does not intersect the x-axis

Thankyou!

Answer
Hi Sherry,
No, its not possible. A quadratic equation has either two real or two imaginary roots. The two real roots could be the same and then we say there is a double root.
The general form of a quadratic equation is
y = ax² + bx + c
and the nature of the roots is always determined by the determinant D, where
D = b² - 4ac
For two real and distinct roots, D > 0
For double roots, D = 0
For complex or imaginary roots, D < 0
Now,
For y = 9x² + 3kx + k
D = (3k)² - 4(9)(k)
 = 9k² - 36k
 = 9k(k-4)

a) There is a double root when
9k(k-4) = 0
i.e k = 0 or 4

b) There are real and distinct roots when
9k(k-4) > 0
i.e k < 0 or k > 4

c) There are complex roots when
9k(k-4) < 0
i.e 0 < k < 4

Regards