Quadratic Equations — Flashcards

Answer

Step 1: Substitute x = 6 into x² - 18x + 72. Step 2: 6² - 18×6 + 72 = 36 - 108 + 72. Step 3: 36 - 108 = -72, and -72 + 72 = 0. Step 4: Since the left side becomes 0, x = 6 is a root. Answer: x = 6 is …

Answer

Step 1: Split the middle term so that the product is 6 and the sum is -5. Step 2: x² - 5x + 6 = (x - 2)(x - 3). Step 3: Use the zero product rule: if a·b = 0, then a = 0 or b = 0. Step 4: x - 2 = 0 or…

Answer

Step 1: Find two numbers whose product is 10 and sum is 7. Step 2: 5 and 2 work. Step 3: x² + 7x + 10 = (x + 5)(x + 2). Step 4: Apply zero product rule. Step 5: x + 5 = 0 or x + 2 = 0. Step 6: x = -5 …

Answer

Step 1: Factorise 2x² + 7x + 3. Step 2: 2x² + 7x + 3 = (2x + 1)(x + 3). Step 3: Apply zero product rule. Step 4: 2x + 1 = 0 or x + 3 = 0. Step 5: x = -1/2 or x = -3. Answer: x = -1/2, -3.

Answer

Use the quadratic formula when factorisation is difficult. For ax² + bx + c = 0, the roots are given by the quadratic formula. Here a = 1, b = 5, c = 6. Substitute into the formula: x = [-5 ± √(5² - 4…

Answer

Step 1: Identify a = 1, b = -4, c = 4. Step 2: Use D = b² - 4ac. Step 3: D = (-4)² - 4×1×4 = 16 - 16 = 0. Step 4: If D = 0, the roots are real and equal. Step 5: Each root is x = -b / 2a = 4 / 2 = 2.

Answer

Step 1: Identify a = 1, b = 2, c = 5. Step 2: D = b² - 4ac = 2² - 4×1×5. Step 3: D = 4 - 20 = -16. Step 4: Since D < 0, the equation has no real root. Answer: No real roots.

Answer

Step 1: a = 2, b = -9, c = 4. Step 2: D = b² - 4ac. Step 3: D = (-9)² - 4×2×4 = 81 - 32 = 49. Step 4: Since D > 0, the roots are real and distinct. Step 5: Distinct real roots occur when b² > 4ac. Ans…