Question
Without solving, determine the nature of roots of the following equations:
- x² − 5x + 6 = 0
- x² − 4x + 4 = 0
- x² − x + 1 = 0
Solution — Step by Step
The discriminant D = b² − 4ac lets us classify roots without actually finding them. This is much faster than solving — especially in MCQs and 1-mark questions.
Why we compute D first: The discriminant is the expression "under the square root" in the quadratic formula. If it's negative, the square root doesn't exist in real numbers. If it's zero, both ± branches give the same value. If it's positive, we get two different real roots.
Step 1: Know the three cases.
Nature of Roots from Discriminant
D = b² − 4ac
D > 0 → Two distinct real roots
D = 0 → Two equal real roots (repeated root)
D < 0 → No real roots (roots are complex/imaginary)
Step 2: Analyse each equation.
Equation 1: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
D = (−5)² − 4(1)(6) = 25 − 24 = 1
D = 1 > 0 → Two distinct real roots.
(They are rational since D is a perfect square: roots are 2 and 3.)
Equation 2: x² − 4x + 4 = 0
a = 1, b = −4, c = 4
D = (−4)² − 4(1)(4) = 16 − 16 = 0
D = 0 → Two equal real roots.
(The repeated root is x = −b/2a = 4/2 = 2.)
Equation 3: x² − x + 1 = 0
a = 1, b = −1, c = 1
D = (−1)² − 4(1)(1) = 1 − 4 = −3
D = −3 < 0 → No real roots. Roots are complex.
📌 Note
Complex roots always come in conjugate pairs. For equation 3, the roots are (1 + i√3)/2 and (1 − i√3)/2, where i = √(−1). For CBSE Class 10, you only need to state "no real roots." For JEE, you may need to find the complex roots.
Why This Works
The discriminant literally tells you about the geometry of the parabola y = ax² + bx + c:
- D > 0: The parabola cuts the x-axis at two points (two real roots).
- D = 0: The parabola just touches the x-axis at one point (equal roots).
- D < 0: The parabola never crosses the x-axis (no real roots).
This geometric interpretation is useful for visualising problems and for solving "for what values of k" questions.
Common JEE Application: Finding k for Equal Roots
A very common question type: "For what value of k does kx² − 3x + 1 = 0 have equal roots?"
For equal roots: D = 0 (−3)² − 4(k)(1) = 0 9 = 4k k = 9/4
💡 Expert Tip
The condition "equal roots" always means D = 0. The condition "no real roots" means D < 0. The condition "real and distinct roots" means D > 0. These three conditions are the backbone of most discriminant questions in JEE Main.
Common Mistake
⚠️ Common Mistake
Mistake: Treating D = 0 as "no solution"
Equal roots means there IS a solution — it's just that both roots are the same value. "No real roots" corresponds to D < 0, not D = 0. When D = 0, x² − 4x + 4 = (x − 2)² = 0 gives x = 2 (repeated). That is a valid, real solution.