How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given #x^2x+6=0#?
1 Answer
Explanation:
The discriminant

If
#Delta > 0# then there are two, distinct, real roots. If#Delta# is a perfect square, then they are both rational. 
If
#Delta = 0# then there is one, repeated, rational, real root. 
If
#Delta < 0# then there are two, distinct, nonReal, Complex roots, which form a complex conjugate pair.
The quadratic equation:
#x^2x+6=0#
is in the form:
#ax^2+bx+c=0#
with
This has discriminant
#Delta = b^24ac = (color(blue)(1))^24(color(blue)(1))(color(blue)(6)) = 124 = 23#
Since
We can find the roots using the quadratic formula:
#x = (b+sqrt(b^24ac))/(2a)#
#color(white)(x) = (b+sqrt(Delta))/(2a)#
#color(white)(x) = (1+sqrt(23))/(2*1)#
#color(white)(x) = (1+sqrt(23)i)/2#
#color(white)(x) = 1/2+sqrt(23)i#