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

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2 Answers

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