Quadratic equation formulas class 11
Answers
Nature of Roots
Let quadratic equation be ax2 + bx + c = 0, whose discriminant is D.
(i) For ax2 + bx + c = 0; a, b , C ∈ R and a ≠ 0, if
(a) D < => Complex roots
(b) D > 0 => Real and distinct roots
(c) D = 0 => Real and equal roots as α = β = – b/2a
(ii) If a, b, C ∈ Q, a ≠ 0, then
(a) If D > 0 and D is a perfect square => Roots are unequal and rational.
(b) If D > 0, a = 1; b, c ∈ I and D is a perfect square. => Roots are integral. .
(c) If D > and D is not a perfect square. => Roots are irrational and unequal.
1. The roots of the quadratic equation: x = (-b ± √D)/2a, where D = b2 – 4ac
2. Nature of roots:
D > 0, roots are real and distinct (unequal)
D = 0, roots are real and equal (coincident)
D < 0, roots are imaginary and unequal
3. The roots (α + iβ), (α – iβ) are the conjugate pair of each other.
4. Sum and Product of roots: If α and β are the roots of a quadratic equation, then
S = α+β= -b/a = coefficient of x/coefficient of x2
P = αβ = c/a = constant term/coefficient of x2
5. Quadratic equation in the form of roots: x2 – (α+β)x + (αβ) = 0
6. The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;
One common root if (b1c2 – b2c1)/(c1a2 – c2a1) = (c1a2 – c2a1)/(a1b2 – a2b1)
Both roots common if a1/a2 = b1/b2 = c1/c2
7. In quadratic equation ax2 + bx + c = 0 or [(x + b/2a)2 – D/4a2]
If a > 0, minimum value = 4ac – b2/4a at x = -b/2a.
If a < 0, maximum value 4ac – b2/4a at x= -b/2a.
8. If α, β, γ are roots of cubic equation ax3 + bx2 + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a
9. A quadratic equation becomes an identity (a, b, c = 0) if the equation is satisfied by more than two numbers i.e. having more than two roots or solutions either real or complex.
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