Let's analyze the Quadratic Formula.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
The discriminant is the expression under the radical sign, b^2-4ac. We know that the discriminant can be used to determine the number of real solutions of a quadratic equation in the following way.
Discriminant
Number of Real Solutions
Negative
None
Zero
One
Positive
Two
We will explain why the discriminant can be used to determine the number of real zeros.
Negative Discriminant
Let's think what would happen if the discriminant is negative, b^2-4ac<0.
x=- b±sqrt(b^2-4ac)/2a
Then we have to calculate a square root of a negative number, which is impossible for real numbers. Therefore, the Quadratic Formula does not make sense in this context. This tells us that the quadratic equation does not have any real zeros.
Zero Discriminant
Now, we will analyze what we get when b^2-4ac= . Let's substitute for the discriminant in the Quadratic Formula.
Finally, let's assume that b^2-4ac>0.
x=- b±sqrt(b^2-4ac)/2a
In this case, we calculate a square root of a positive number. Each positive real number has exactly one real root. This means that the solutions for the equation are x= - b±sqrt(b^2-4ac)2a. Let's separate them into the positive and negative cases.
x=- b±sqrt(b^2-4ac)/2a
x_1=- b-sqrt(b^2-4ac)/2a
x_2=- b+sqrt(b^2-4ac)/2a
Therefore, we have exactly two real solutions, x_1 and x_2.
