Where the roots go when D turns negative
"Where the roots go when D turns negative" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Quadratics with a negative discriminant" in Complex Numbers and Quadratic Equations. Slide c in x² − 4x + c = 0. Left: the parabola y = x² − 4x + c and its x-intercepts. Right: the two roots as points of the Argand plane. D > 0: two real roots. D = 0: one repeated root. D < 0: two conjugate roots −b/2a ± i√(−D)/2a, and the real part −b/2a is where the parabola has its vertex. JEE Main asked 79 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Quadratics with a negative discriminant in Complex Numbers and Quadratic Equations. Change the values and watch what happens.
In JEE Main
42 shifts analysed
79 questions from Complex Numbers and Quadratic Equations across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §4.1
- The roots of ax² + bx + c = 0, with a, b, c real and a ≠ 0, are x = (−b ± √D)/2a, where D = b² − 4ac.
- When D < 0, √D = √(4ac − b²) i, so the roots are x = (−b ± √(4ac − b²) i)/2a. They are two non-real numbers with the same real part −b/2a and opposite imaginary parts: a conjugate pair.
- Worked example: x² − 4x + 5 = 0 has D = 16 − 20 = −4, so √D = 2i and x = (4 ± 2i)/2 = 2 ± i. Check: (2 + i)² − 4(2 + i) + 5 = 3 + 4i − 8 − 4i + 5 = 0.
- Worked example: x² + x + 1 = 0 has D = 1 − 4 = −3, so x = (−1 ± √3 i)/2.
- Worked example: x² + 9 = 0 gives x² = −9 and x = ±3i; in general x² + k² = 0 has the roots ±ki.
- The relations sum of roots = −b/a and product of roots = c/a still hold: for 2 ± i the sum is 4 and the product is (2 + i)(2 − i) = 4 + 1 = 5, matching −(−4)/1 and 5/1.
- On a graph, a > 0 with D < 0 means the parabola y = ax² + bx + c stays above the x-axis. Its vertex has x = −b/2a, which is exactly the real part of the two complex roots.
Questions
3 answers
- What does the "Where the roots go when D turns negative" simulation show?
- Slide c in x² − 4x + c = 0. Left: the parabola y = x² − 4x + c and its x-intercepts. Right: the two roots as points of the Argand plane.
- Does JEE Main ask questions from Complex Numbers and Quadratic Equations?
- Yes. JEE Main asked 79 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Quadratics with a negative discriminant" in Complex Numbers and Quadratic Equations (Class 11 Maths).
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