Quadratic%20Equations

Answer :

2x² + 5x – k = 0

                   Δ = b² – 4ac = 81

                             ∴ 25 + (4 × 2 × k) = 81

                             ∴ 8k = 56

                             ∴ k = 7

Answer :

Discriminate = Δ = b² – 4ac

                   Here b = – 3 √3, a = √5, c = – 2√5

                             ∴ Δ = (– 3 √3)² – 4(√5) (– 2 √5)

                                   = 27 + 40

                               Δ = 67

Answer :

kx² + 1 = kx + 3x – 11x²

                   ∴ (k + 11)x² – (k + 3)x + 1 = 0

                   ∵ Δ = b² – 4ac

                           Δ = [– (k + 3)]² – 4(k + 11)(1)

                             = k² + 6k + 9 – 4k – 44

                             = k² + 2k – 35

                   Roots are real and equal

                             ∴ Δ ≥ 0

                             ∴ k² + 2k – 35 ≥ 0

                             ∴ k² + 2k + 1 ≥ 36

                                      |k + 1|² ≥ 36

                                      |k + 1| ≥ 6

                             ∴ – 6 ≥ (k + 1) ≥ 6

                             ∴ – 7 ≥ k ≥ 5

                   i.e. k = – 7 and k = 5

                             ∴ k ∈ {– 7, 5}

Answer :

          ax² + bx + c = 0

                   Let α, β are roots of equation

                                        α + β         = {(– b ) / a}         αβ = (c/a)

                                        α + β         = (1 / α²) + (1 / β²)

                                      ∴ α + β        = [{(α + β)² – 2αβ} / {α² β²}]

                                ∵ {(– b) / a}       = [({(– b) / a}² – 2 (c/a)) / (c / a)²]

                                ∴ {(– b) / a}       = [{(b² / a²) – (2c / a)} / {c² / a²⁰}]

                                ∴ {(– b) / a}       = [(b² – 2ac) / c²]

                                      ∴ – bc²         = ab² – 2a²c

                                      ∴ 2a²c         = ab² + bc²

                                          a²c = [(ab² + bc²) / 2]

                             ∴ ab², a²c and bc² are in arithmetic progression.

Answer :

x² – m(2x – 8) – 15 = 0

                             ∴ x² – 2mx + 8m – 15 = 0

                   roots are equal hence Δ = 0

                             ∴ b² – 4 ac = 0

                             ∴ (– 2m)² – 4(1) (8m – 15) = 0

                             ∴ 4m² – 32m + 60 = 0

                             ∴ m = 5, m = 3