2
&Quadratic equation: && ax^2+ bx+ c=0
&Solutions:&&x=- b±sqrt(b^2-4 a c)/2 a
&Discriminant:&& b^2-4 a c
The table below summarizes the relationship between the discriminant and the number of real solutions.
Discriminant (b^2-4ac)
Number of Real Solutions
Positive
2
Zero
1
Negative
0
Let's make some observations about the terms in the discriminant.
The coefficient b is real, so b^2 is positive or zero.
It is given that a and c are different signs, so their product a c is negative.
Subtracting a negative number from a non-negative number results in a positive number.
b^2-4 a c>0
If a and c are different signs, then
the discriminant is positive.
It is always true that if a and c are different signs, then the two solutions are both real.
b Let's recall the Quadratic Formula that gives the solutions of a quadratic equation.
2
&Quadratic equation: && ax^2+ bx+ c=0
&Solutions:&&x=- b±sqrt(b^2-4 a c)/2 a
&Discriminant:&& b^2-4 a c
If the coefficients are integer numbers, then the nature of the solutions depends on the discriminant.
Discriminant (b^2-4ac)
Solutions
Complete square
Rational
Not-complete square
Irrational
If we only know that the discriminant is greater than 1, then it may or may not be a complete square.
Let's see two examples.
Equation
Discriminant
Quadratic Formula
Solutions
x^2+4x+3=0
4^2-4(1)(3)=4
-4±sqrt(4)/2
-3, -1
x^2-3=0
0^2-4(1)(-3)=12
0±sqrt(12)/2
-sqrt(3), sqrt(3)
It is sometimes true that the roots are real irrational numbers when the discriminant is greater than 1.
