JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
QUADRATIC EQUATIONS
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JEE MATH APEX
1. General Form & Roots
📌 Standard Quadratic Equation & Quadratic Formula
• General form: ax² + bx + c = 0 (where a ≠ 0)
• Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
• Roots: α = [−b + √D] / 2a and β = [−b − √D] / 2a
• Discriminant: D = b² − 4ac
• Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
• Roots: α = [−b + √D] / 2a and β = [−b − √D] / 2a
• Discriminant: D = b² − 4ac
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2. Nature of Roots
📌 Conditions Based on Discriminant (D = b² − 4ac)
• If D > 0: Roots are real and distinct
• If D = 0: Roots are real and equal (coincident)
• If D < 0: Roots are imaginary (complex conjugate)
• If D ≥ 0: Roots are real
• If D is a perfect square and a, b, c are rational: Roots are rational
• If D > 0 but not a perfect square: Roots are irrational (occur in conjugate pairs)
• If D = 0: Roots are real and equal (coincident)
• If D < 0: Roots are imaginary (complex conjugate)
• If D ≥ 0: Roots are real
• If D is a perfect square and a, b, c are rational: Roots are rational
• If D > 0 but not a perfect square: Roots are irrational (occur in conjugate pairs)
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3. Sum & Product of Roots
📌 Vieta’s Formulas
• Sum of roots: α + β = −b/a
• Product of roots: αβ = c/a
• Difference of roots: |α − β| = √D / |a|
• Sum of reciprocals: 1/α + 1/β = −b/c
• α² + β² = (α + β)² − 2αβ = b²/a² − 2c/a
• Product of roots: αβ = c/a
• Difference of roots: |α − β| = √D / |a|
• Sum of reciprocals: 1/α + 1/β = −b/c
• α² + β² = (α + β)² − 2αβ = b²/a² − 2c/a
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4. Formation of Quadratic Equation
📌 From Given Roots
• If roots are α and β, equation is: x² − (α + β)x + αβ = 0
• In general: x² − (Sum of roots)x + (Product of roots) = 0
• If roots are reciprocal of each other: a = c
• If roots are negative of each other: b = 0
• If one root is zero: c = 0
• If both roots are zero: b = 0, c = 0
• In general: x² − (Sum of roots)x + (Product of roots) = 0
• If roots are reciprocal of each other: a = c
• If roots are negative of each other: b = 0
• If one root is zero: c = 0
• If both roots are zero: b = 0, c = 0
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5. Common Roots of Two Quadratic Equations
📌 Condition for Common Roots
• Given: a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0
One common root:
• (c₁a₂ − c₂a₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁)
• Common root: x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
Both roots common:
• a₁/a₂ = b₁/b₂ = c₁/c₂
One common root:
• (c₁a₂ − c₂a₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁)
• Common root: x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
Both roots common:
• a₁/a₂ = b₁/b₂ = c₁/c₂
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6. Symmetric Functions of Roots
📌 Expressions in Terms of α+β and αβ
• α² + β² = (α + β)² − 2αβ
• α³ + β³ = (α + β)³ − 3αβ(α + β)
• α² − β² = (α + β)(α − β) = (α + β)√[(α + β)² − 4αβ]
• α⁴ + β⁴ = [(α + β)² − 2αβ]² − 2(αβ)²
• (α − β)² = (α + β)² − 4αβ = D/a²
• α/β + β/α = (α² + β²)/αβ = [(α+β)² − 2αβ]/αβ
• α³ + β³ = (α + β)³ − 3αβ(α + β)
• α² − β² = (α + β)(α − β) = (α + β)√[(α + β)² − 4αβ]
• α⁴ + β⁴ = [(α + β)² − 2αβ]² − 2(αβ)²
• (α − β)² = (α + β)² − 4αβ = D/a²
• α/β + β/α = (α² + β²)/αβ = [(α+β)² − 2αβ]/αβ
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7. Transformation of Roots
📌 New Equation from Transformed Roots
• Roots increased by k (α+k, β+k): a(x−k)² + b(x−k) + c = 0
• Roots decreased by k (α−k, β−k): a(x+k)² + b(x+k) + c = 0
• Roots multiplied by k (kα, kβ): a(x/k)² + b(x/k) + c = 0 → ax² + kbx + k²c = 0
• Reciprocal roots (1/α, 1/β): cx² + bx + a = 0
• Roots squared (α², β²): a²x² − (b² − 2ac)x + c² = 0
• Roots decreased by k (α−k, β−k): a(x+k)² + b(x+k) + c = 0
• Roots multiplied by k (kα, kβ): a(x/k)² + b(x/k) + c = 0 → ax² + kbx + k²c = 0
• Reciprocal roots (1/α, 1/β): cx² + bx + a = 0
• Roots squared (α², β²): a²x² − (b² − 2ac)x + c² = 0
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8. Sign of Quadratic Expression
📌 Conditions for Positive/Negative Values
• ax² + bx + c > 0 ∀ x ∈ R ⇔ a > 0 and D < 0
• ax² + bx + c < 0 ∀ x ∈ R ⇔ a < 0 and D < 0
• ax² + bx + c ≥ 0 ∀ x ∈ R ⇔ a > 0 and D ≤ 0
• ax² + bx + c ≤ 0 ∀ x ∈ R ⇔ a < 0 and D ≤ 0
• ax² + bx + c < 0 ∀ x ∈ R ⇔ a < 0 and D < 0
• ax² + bx + c ≥ 0 ∀ x ∈ R ⇔ a > 0 and D ≤ 0
• ax² + bx + c ≤ 0 ∀ x ∈ R ⇔ a < 0 and D ≤ 0
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9. Location of Roots
📌 Roots Relative to a Real Number k
• Both roots greater than k: D ≥ 0, −b/2a > k, a·f(k) > 0
• Both roots less than k: D ≥ 0, −b/2a 0
• k lies between roots: a·f(k) < 0
• Exactly one root in (k₁, k₂): f(k₁) · f(k₂) < 0
• Both roots in (k₁, k₂): D ≥ 0, k₁ < −b/2a 0, a·f(k₂) > 0
• Both roots less than k: D ≥ 0, −b/2a 0
• k lies between roots: a·f(k) < 0
• Exactly one root in (k₁, k₂): f(k₁) · f(k₂) < 0
• Both roots in (k₁, k₂): D ≥ 0, k₁ < −b/2a 0, a·f(k₂) > 0
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10. Maximum & Minimum Value
📌 Vertex of Parabola
• Vertex: x = −b/2a, y = −D/4a
• If a > 0: Minimum value = −D/4a at x = −b/2a
• If a < 0: Maximum value = −D/4a at x = −b/2a
• Range when a > 0: [−D/4a, ∞)
• Range when a < 0: (−∞, −D/4a]
• If a > 0: Minimum value = −D/4a at x = −b/2a
• If a < 0: Maximum value = −D/4a at x = −b/2a
• Range when a > 0: [−D/4a, ∞)
• Range when a < 0: (−∞, −D/4a]
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11. Cubic & Higher Degree (Advanced)
📌 Cubic Equation ax³ + bx² + cx + d = 0
• Sum of roots: α + β + γ = −b/a
• Sum of product of roots taken two at a time: αβ + βγ + γα = c/a
• Product of roots: αβγ = −d/a
• For bi-quadratic ax⁴ + bx³ + cx² + dx + e = 0:
Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a
• Sum of product of roots taken two at a time: αβ + βγ + γα = c/a
• Product of roots: αβγ = −d/a
• For bi-quadratic ax⁴ + bx³ + cx² + dx + e = 0:
Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a
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12. Special Equations & Substitutions
📌 Equations Reducible to Quadratic
• ax⁴ + bx² + c = 0 → Put x² = t → at² + bt + c = 0
• a·(x + 1/x)² + b·(x + 1/x) + c = 0 → Put x + 1/x = t
• a·√x + b·x + c = 0 → Put √x = t
• Exponential: a·p2x + b·px + c = 0 → Put px = t
• a·(x + 1/x)² + b·(x + 1/x) + c = 0 → Put x + 1/x = t
• a·√x + b·x + c = 0 → Put √x = t
• Exponential: a·p2x + b·px + c = 0 → Put px = t
✅ QUADRATIC EQUATIONS ✅
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— JEE MATH APEX —
JEE MATH APEX 
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