Question

Use the discriminant to determine the number and type of solutions for the given equation x^2+2x+6=0

Answer 1

The discriminant is the portion of the quadratic formula under the root sign: b²-4ac

A quadratic equation in standard form is ax²+bx+c=0. In your equation:
a=1, b=2, and c=6. Substituting them into the equation:

2²-4(1)(6) 
=4-24
=(-20)

When the discriminant is negative there are two, imaginary solutions.

If the discriminant equals a positive, perfect square number (like 4, 9, 16, etc), there will be two, rational roots.
If the discriminant equals a positive, nonperfect square number (like 3, 15, 23, etc), there will be two, irrational roots.
If the discriminant equals 0, there will be one double root 

A quadratic equation graphs a parabola. The graph of a quadratic with a  positive discriminant will pass through the x-axis twice. If the discriminant is 0, it will touch the x-axis and 'bounce' back the way it came. If the discriminant is negative, it will not touch the x-axis.