01 · THE DISCRIMINANT
What is the Discriminant (D)?
In the quadratic formula x = (-b ± √(b² - 4ac)) / (2a), the expression under the square root sign b² - 4ac is called the Discriminant, denoted by D.
D = b² - 4acThe value of D determines whether the equation has real, equal, or no real roots!
02 · THREE CONDITIONS
The 3 Cases for Nature of Roots
01
If D > 0 (Positive)
Two distinct real roots exist: x = (-b ± √D) / 2a02
If D = 0 (Zero)
Two equal real roots exist: x = -b / 2a03
If D < 0 (Negative)
No real roots exist (square root of negative number is not real).03 · SOLVED EXAMPLES
Determining Nature of Roots
EXAMPLE 1
Find the discriminant of 2x² - 4x + 3 = 0 and state the nature of its roots
Step 1: Identify coefficients: a = 2, b = -4, c = 3.
Step 2: Calculate D = b² - 4ac:
D = (-4)² - 4(2)(3) = 16 - 24 = -8
Since D = -8 < 0, the given equation has NO real roots.
EXAMPLE 2
Find the value of k if 2x² + kx + 3 = 0 has two equal real roots
Step 1: Condition for equal roots is D = 0.
Here, a = 2, b = k, c = 3.
b² - 4ac = 0
k² - 4(2)(3) = 0
k² - 24 = 0 ⟹ k² = 24
k = ± √24 = ± 2√6
k² - 4(2)(3) = 0
k² - 24 = 0 ⟹ k² = 24
k = ± √24 = ± 2√6
04 · QUICK SUMMARY
Summary Table
D > 0
Two distinct real roots
D = 0
Two equal real roots
D < 0
No real roots