Discriminant of a Quadratic Equation
What is the discriminant?
The discriminant of a quadratic equation is a simple expression that tells us whether the equation has real solutions. It is denoted by the uppercase Greek letter delta. $$ \Delta = b^2-4ac $$ The letters a, b and c represent the coefficients of the quadratic equation $$ ax^2 + bx + c = 0$$
What is it used for?
The sign of the discriminant allows us to determine the nature of the solutions:
- Δ>0
If the discriminant is positive (Δ>0), the equation has two distinct real solutions. - Δ=0
If the discriminant equals zero (Δ=0), the equation has one real solution, meaning the two solutions coincide. - Δ<0
If the discriminant is negative (Δ<0), the equation has no solutions in the set of real numbers (R).
Note. It is called the discriminant because it “helps to distinguish” between different types of quadratic equations before actually solving them.
A practical example
Example 1
Consider the equation:
$$ x^2 - 3x - 4 = 0 $$
The coefficients are a=1, b=-3, c=-4.
The discriminant is:
$$ \Delta = b^2-4ac = (-3)^2 -4 \cdot 1 \cdot (-4) = 9 + 16 = 25 $$
Since the discriminant is positive, the equation has two distinct real solutions.
Note. I still do not know the actual values of the solutions, but I already know that two real solutions exist. To find them, I simply apply the quadratic formula.
Example 2
Consider the equation:
$$ x^2 - 4x + 4 = 0 $$
The coefficients are a=1, b=-4, c=4.
The discriminant is:
$$ \Delta = b^2-4ac = (-4)^2 -4 \cdot 1 \cdot (4) = 16 - 16 = 0 $$
Since the discriminant is zero, the equation has exactly one real solution.
Note. There are technically two solutions, but they are identical. This is known as a double root. Once again, we can find it by applying the quadratic formula.
Example 3
Consider the equation:
$$ x^2 - 2x + 4 = 0 $$
The coefficients are a=1, b=-2, c=4.
The discriminant is:
$$ \Delta = b^2-4ac = (-2)^2 -4 \cdot 1 \cdot 4 = 4 - 16 = -12 $$
Since the discriminant is negative, the equation has no real solutions.
Note. The fact that there are no real solutions does not mean the equation has no solutions at all. For instance, it may still have solutions in the set of complex numbers. That is a different topic.
And so on.
