Cubic Equations and Nature of Roots
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Direct answer
A cubic ax^3 + bx^2 + cx + d = 0 with a ≠ 0 always has at least one real root, and its two turning points decide whether there are three. If the roots are α, β, γ, then α + β + γ = -b/a, αβ + βγ + γα = c/a and αβγ = -d/a. The discriminant Δ = 18abcd - 4b^3 d + b^2 c^2 - 4ac^3 - 27a^2 d^2 reads as: Δ > 0 gives three distinct real roots, Δ = 0 a repeated root, and Δ < 0 one real plus two complex conjugate roots. Shifting x = t - b/(3a) strips the quadratic term, giving the depressed cubic t^3 + pt + q whose compact discriminant is -4p^3 - 27q^2.
What you must remember
- Vieta triples: sum -b/a, pairwise sum c/a, product -d/a; α^2 + β^2 + γ^2 = (α + β + γ)^2 - 2(αβ + βγ + γα).
- Turning-point test: critical points solve 3ax^2 + 2bx + c = 0; if none is real, the cubic is monotonic with exactly one real root; if two exist (x1 < x2), three distinct real roots need f(x1)·f(x2) < 0, and a double root appears when one of these values is zero.
- Roots in AP: holds iff 2b^3 - 9abc + 27a^2 d = 0, the middle root then being -b/(3a).
- Roots in GP: middle root β = -c/b with the condition ac^3 = b^3d; sanity check with roots 1, 2, 4: the polynomial x^3 - 7x^2 + 14x - 8 satisfies 1 × 14^3 = (-7)^3 × (-8) = 2744.
- Derivative condition b^2 - 3ac > 0 is necessary for three real roots (real turning points must exist) but not sufficient — their values must straddle zero.
- Complex roots arrive as conjugate pairs for real coefficients, so the real-root count is 3 or 1, never 2 or 0.
Working a parameter range
The model problem: find every k for which x^3 - 3x + k = 0 has three distinct real roots. Differentiate: f'(x) = 3x^2 - 3, zero at x = -1 and x = 1. Because the leading coefficient is positive, x = -1 is the local maximum and x = 1 the local minimum. Evaluate f(-1) = 2 + k and f(1) = k - 2. Three distinct real roots need the maximum above the axis and the minimum below it: (2 + k)(k - 2) < 0, so -2 < k < 2. The endpoints behave exactly as expected — k = 2 factors as (x - 1)^2(x + 2) with the minimum touching the axis, and k = -2 as (x + 1)^2(x - 2). Cross-check with the depressed form (this cubic is already depressed: p = -3, q = k): -4p^3 - 27q^2 > 0 gives 108 - 27k^2 > 0, the same band. Graph logic and discriminant agreeing is what a full-marks solution shows.
How the exam frames it
JEE Main keeps cubics as fast Vieta arithmetic — numerical-value questions on α^3 + β^3 + γ^3 or α^2 + β^2 + γ^2 that reward clean expansion — plus the occasional root-count question read off a turning-point sketch. Advanced prefers the parameter-range format above, or cubics camouflaged inside functional equations and divisibility arguments. The classic trap is quoting the AP condition as sufficient for three real roots: the roots i, 0, -i form a genuine arithmetic progression and satisfy x^3 + x = 0 (2b^3 - 9abc + 27a^2 d = 0 holds), yet only 0 is real — always re-check the discriminant. Cardano's formula itself sits outside the syllabus and has never been required; the turning-point method replaces it entirely. Both exams draw this from the standard equations-and-inequalities pocket of the Mathematics syllabus.
Frequently asked questions
What is the condition for the roots of ax^3 + bx^2 + cx + d = 0 to be in AP?
2b^3 - 9abc + 27a^2 d = 0, with the middle root equal to -b/(3a); add a discriminant check to keep all three roots real.
How many real roots can a cubic with real coefficients have?
Three or one — never two or zero, because non-real roots occur only as conjugate pairs and one sign change across the graph is guaranteed.
When does a cubic have a repeated root?
When f and f' share a common root, equivalently when the discriminant vanishes; the repeated root then satisfies both f(x) = 0 and 3ax^2 + 2bx + c = 0.
For which k does x^3 - 3x + k = 0 have exactly one real root?
k < -2 or k > 2, the region where both turning-point values sit on the same side of the axis.
What does the substitution x = t - b/(3a) achieve?
It removes the quadratic term, producing the depressed cubic t^3 + pt + q, whose compact discriminant -4p^3 - 27q^2 instantly decides between one and three real roots.