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Quadratic Equations · Class 11 Maths Notes
Class 11 notes on Quadratic Equations: the discriminant and nature of roots, Vieta's sum and product relations, the parabola view for range and sign, and common roots.
In short
The discriminant tells you what the roots look like before you find them: positive and it has two distinct real roots, zero and they coincide, negative and they are a complex conjugate pair. Vieta's relations then give their sum and product without solving, and the parabola answers every question about sign and range at a glance.
Contents
The Quadratic Equation and Its Roots
A quadratic equation is any equation of the form ax² + bx + c = 0 with a ≠ 0. Its two solutions, the roots, are where the parabola y = ax² + bx + c crosses the x-axis. Almost everything in this chapter flows from one formula.
x = [ −b ± √(b² − 4ac) ] / 2a
The two roots come from completing the square on the general equation. The expression under the root, b² − 4ac, is called the discriminant and is written D or Δ. It alone decides the nature of the roots, which is why it is the single most important quantity in the chapter.
The Discriminant: Nature of Roots
Before solving, the discriminant tells you what kind of roots to expect. This is a guaranteed exam question.
| Discriminant D = b² − 4ac | Nature of roots |
|---|---|
| D > 0 (and a perfect square) | Real, distinct, rational |
| D > 0 (not a perfect square) | Real, distinct, irrational (conjugate surds) |
| D = 0 | Real and equal (one repeated root, −b/2a) |
| D < 0 | No real roots (complex conjugate pair) |
For x² − 4x + 4 = 0: D = (−4)² − 4(1)(4) = 16 − 16 = 0, so the roots are real and equal (x = 2, twice). For x² + x + 1 = 0: D = 1 − 4 = −3 < 0, so the roots are a complex conjugate pair with no real value.
If a quadratic with real coefficients has one complex root p + qi, its other root must be the conjugate p − qi. Complex roots never appear alone in a real-coefficient equation. The same pairing holds for irrational surd roots like 2 + √3 and 2 − √3.
Sum and Product of Roots (Vieta's Relations)
You rarely need the actual roots. Their sum and product come straight from the coefficients, and this shortcut solves a huge fraction of JEE quadratic questions.
sum α + β = −b/a
product αβ = c/a
Reversing the relations, any quadratic can be written from its roots:
x² − (α + β)x + αβ = 0
This is the fastest way to construct an equation with required roots, or one whose roots are transformed versions of another's.
For x² − 5x + 6 = 0: sum of roots = 5, product = 6. Without solving, we know the roots multiply to 6 and add to 5, so they are 2 and 3. Check: 2 + 3 = 5, 2 × 3 = 6.
Many questions ask for a symmetric function of the roots without the roots themselves. Build them from the sum S = α + β and product P = αβ:
• α² + β² = S² − 2P
• (α − β)² = S² − 4P
• α³ + β³ = S³ − 3PS
• 1/α + 1/β = S/P
The Graph, Vertex and Range
Seeing the quadratic as a parabola answers every maximum, minimum and range question at a glance.
The parabola y = ax² + bx + c has its vertex at x = −b/2a. If a > 0 it opens upward and the vertex is a minimum; if a < 0 it opens downward and the vertex is a maximum. The extreme value is found by substituting x = −b/2a, and equals (4ac − b²)/4a, that is −D/4a.
Find the least value of f(x) = x² − 4x + 7. Here a = 1 > 0, so there is a minimum at x = −(−4)/2 = 2. Then f(2) = 4 − 8 + 7 = 3. The range is therefore [3, ∞).
A quadratic with a > 0 and D < 0 is positive for every real x (the parabola sits entirely above the axis). This fact is the key to many always-positive and inequality problems: check the leading sign and the discriminant together.
Common Roots and Special Cases
A few standard set-ups round out the chapter and appear regularly in JEE Main.
If two quadratics a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 share one common root, then (a₁c₂ − a₂c₁)² = (b₁c₂ − b₂c₁)(a₁b₂ − a₂b₁). If both roots are common, the coefficients are proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.
• Roots are reciprocals (αβ = 1) when c = a.
• Roots are equal in magnitude, opposite in sign (α + β = 0) when b = 0.
• One root is zero when c = 0.
• Both roots positive: need D ≥ 0, sum > 0 and product > 0.
For what k does x² − 6x + k = 0 have equal roots? Set D = 0: (−6)² − 4(1)(k) = 0, so 36 − 4k = 0, giving k = 9. The repeated root is then x = −b/2a = 3.
Quadratic Equations: One-Page Formula Sheet
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