Quadratic Quest

An exciting adventure through the world of quadratic equations!

Problem Set 1: Nature of Roots

Determine the nature of the roots for the following quadratic equations by calculating the discriminant.

Problem (i): 15x² + 11x + 2 = 0
15x² + 11x + 2 = 0
Step 1: Identify coefficients: a = 15, b = 11, c = 2
Step 2: Calculate discriminant (D) = b² - 4ac
Step 3: D = (11)² - 4 × 15 × 2 = 121 - 120 = 1
Step 4: Since D > 0 and is a perfect square, roots are real, distinct, and rational.
Problem (ii): x² - x - 1 = 0
x² - x - 1 = 0
Step 1: Identify coefficients: a = 1, b = -1, c = -1
Step 2: Calculate discriminant (D) = b² - 4ac
Step 3: D = (-1)² - 4 × 1 × -1 = 1 + 4 = 5
Step 4: Since D > 0 but not a perfect square, roots are real, distinct, and irrational.
Problem (v): 9a²b²x² - 24abcdx + 16c²d² = 0
9a²b²x² - 24abcdx + 16c²d² = 0 (a ≠ 0, b ≠ 0)
Step 1: Identify coefficients: a = 9a²b², b = -24abcd, c = 16c²d²
Step 2: Calculate discriminant (D) = b² - 4ac
Step 3: D = (-24abcd)² - 4 × 9a²b² × 16c²d² = 576a²b²c²d² - 576a²b²c²d² = 0
Step 4: Since D = 0, roots are real and equal.

Problem Set 2: Find 'k' for Equal Roots

Find the value(s) of 'k' for which the roots of the following equations are real and equal.

Problem (i): (5k - 6)x² + 2kx + 1 = 0
(5k - 6)x² + 2kx + 1 = 0
Step 1: For equal roots, discriminant D = 0
Step 2: Identify coefficients: a = (5k - 6), b = 2k, c = 1
Step 3: D = b² - 4ac = (2k)² - 4 × (5k - 6) × 1 = 4k² - 20k + 24
Step 4: Set D = 0: 4k² - 20k + 24 = 0
Step 5: Simplify: k² - 5k + 6 = 0
Step 6: Solve: (k - 2)(k - 3) = 0 ⇒ k = 2 or 3
Step 7: Verify a ≠ 0: For k=2, a=4; for k=3, a=9. Both valid.

Challenge Problems

Problem 3: Arithmetic Progression Proof
(a - b)x² + (b - c)x + (c - a) = 0
Step 1: For equal roots, discriminant D = 0
Step 2: D = (b - c)² - 4 × (a - b) × (c - a) = 0
Step 3: Expand: b² - 2bc + c² - 4(ac - a² - bc + ab) = 0
Step 4: Simplify: b² - 2bc + c² - 4ac + 4a² + 4bc - 4ab = 0
Step 5: Combine like terms: 4a² + b² + c² - 4ab - 4ac + 2bc = 0
Step 6: Recognize as perfect square: (2a - b - c)² = 0 ⇒ 2a = b + c
Step 7: This means b, a, c are in arithmetic progression (since a - b = c - a).