Algebra — Important Questions

ax² + bx + c = 0, where a ≠ 0

Step 1: A quadratic equation is a polynomial equation of degree 2. Step 2: The general form is ax² + bx + c = 0. Step 3: The condition a ≠ 0 is essential — if a = 0, the equation becomes linear (ax + b = 0), not quadratic. Step 4: Options B and C are linear and invalid quadratic forms respectively. Option D is cubic. Common mistake: Forgetting the condition a ≠ 0.

x = 2 and x = 3

Step 1: Here a = 1, b = -5, c = 6. Step 2: Discriminant Δ = b² - 4ac = 25 - 24 = 1. Step 3: x = (-b ± √Δ)/(2a) = (5 ± 1)/2. Step 4: x = 6/2 = 3 or x = 4/2 = 2. Verification: (2-2)(2-3) = 0 ✓ and (3-2)(3-3) = 0 ✓. Common mistake: Forgetting to take both + and - cases.

-3/2

Step 1: For ax² + bx + c = 0, sum of roots = -b/a. Step 2: Here a = 2, b = 3. Step 3: Sum of roots = -b/a = -3/2. Step 4: No further calculation needed. Common mistake: Computing b/a = 3/2 (forgetting the negative sign) or confusing sum and product formulas.

12

Step 1: For ax² + bx + c = 0, product of roots = c/a. Step 2: Here a = 1, b = -7, c = 12. Step 3: Product of roots = c/a = 12/1 = 12. Step 4: Verification: roots are 3 and 4, and 3 × 4 = 12 ✓. Common mistake: Using -c/a or confusing product with sum formula.