Quadratic Equations

Answer

Step 1: Identify values that multiply to 6 and add to 5 → these are 2 and 3 Step 2: Write the equation as (x + 2)(x + 3) = 0 Step 3: Set each factor to zero → x + 2 = 0 OR x + 3 = 0 Step 4: Solve each…

Answer

Step 1: Multiply a × c = 2 × 1 = 2 Step 2: Find two numbers that multiply to 2 and add to -3 → these are -2 and -1 Step 3: Rewrite middle term: 2x² - 2x - x + 1 = 0 Step 4: Factor by grouping: 2x(x - …

Answer

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Identifying quadratic equations: (a) 2x + 3 = 0 → NOT quadratic (degree 1, no x² term) (b) …

Answer

Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a Where: - a = coefficient of x² - b = coefficient of x - c = constant term - b² - 4ac is called the discriminant Example: Solve x² + 5x + 6 = 0 Here a =…

Answer

Given: x² - 4x + 3 = 0 a = 1, b = -4, c = 3 Step 1: Calculate discriminant = b² - 4ac = (-4)² - 4(1)(3) = 16 - 12 = 4 Step 2: Apply formula x = [-b ± √(b² - 4ac)] / 2a Step 3: x = [4 ± √4] / 2 = [4 ±…

Answer

The discriminant (D or Δ) of a quadratic equation ax² + bx + c = 0 is: D = b² - 4ac It tells us about the NATURE of roots: 1. If D > 0 (discriminant is positive) → Two distinct real roots Example…

Answer

Step 1: Identify a = 2, b = -8, c = 8 Step 2: Calculate discriminant D = b² - 4ac D = (-8)² - 4(2)(8) = 64 - 64 = 0 Step 3: Since D = 0, the equation has TWO EQUAL REAL ROOTS Step 4: Find the roots …

Answer

To check if x = 3 is a root, substitute it into the equation: Step 1: Replace x with 3 in 2x² - x - 15 = 0 Step 2: 2(3)² - (3) - 15 = 2(9) - 3 - 15 Step 3: = 18 - 3 - 15 = 0 Since LHS = 0 = RHS, YES…