How can we determine the number and type of solutions to a quadratic equation without actually solving it?

Question

 I just need to know for my College Math class. I feel like I am not doing so great in class and need HELP.

Answer 1

In reality if it is a quadratic then there are always two roots.  The two roots may be the same number, but still there are two.  What I think your class is working on is the discrimanant.  That's the (b^2-4ac) term under the radical.  If this term is zero (i.e., in X^2 + 2X + 1) then the two roots of the equation are the same number and the quadratic is said to have only one solution.  If the value of (b^2-4ac) is a positive then you will have two unique real roots by using the + value of its square root and by using its - negative value in the equation

 

 

      -b + (b^2-4ac)^0.5

X = ________________

 

                2a

 

If the value of (b^2-4ac) is negative then you end up with two roots that are both imaginary.

Answer 2

If you look at the quadratic formula for the solution of a²+bx+c=0 where for this case I assume a,b,c are all real numbers   x=(-b±√(b²-4ac))/2a

The term b²-4ac, called the discriminant, gives the answers

If it is 0 you have a double root x=-b/2a

If it is positive, then two real roots x=(-b±√(b²-4ac))/2a

If it is negative then two complex conjugate roots x=(-b±i√(4ac-b²))/2a