Question
1.3wx = 40 – wy 2. b2 = 2b + p
3. d2 + d = q Now, observe the given conditions: One root of equation 1 is -8. There exists a common root between equations 1 and 2, and its value is greater than zero. There exists a common root between equations 2 and 3. The difference between the roots of equation 3 is less than 6. All the roots are real numbers and integers. Below are three quadratic equations numbered 1, 2, and 3. Examine the data attentively and calculate the sum of the roots of all three equations where the roots are greater than the other root.Solution
ATQ, Equation 1, 3wx = 40 - wy Case 1) x = 1, y = 2 3w = 40 - w2 w2 + 3w - 40 = 0 At w = - 8 (- 8)3 + 3 × (- 8) - 40 = 0 64 - 24 - 40 = 0 0 = 0 LHS = RHS Other root of Eq. 1 = (- 40)/(- 8) = 5 Greater root = 5 Case 2) x = 2, y = 1 3w2 = 40 - a 3w2 + w - 40 = 0 At w = - 8 3 × (- 8)2 + (- 8) - 40 = 0 3 × 64 - 8 - 40 = 0 192 - 48 = 0 144 ≠ 0 Equation 2, b2 = 2b + p b2 - 2b - p = 0 1 root of equation = 5 (common root of equation 1 & 2 and greater than 0) Other root = 2 - 5 = - 3- p = 5 × (- 3) p = 15 Greater root = 5 Equation 3, d2 + d = q d2 + d - q = 0 If common root of equation 2 & 3 is 5, then other root of equation 3 = (- 1) - 5 = - 6 Difference = 5 - (- 6) = 11 > 6 (Not possible) If common root of equation 2 & 3 is - 3, then other root of equation 3 = (- 1) - (- 3)= 2 Difference = 2 - (- 3) = 5 < 6 (Possible) Root of equation 3 = 2, - 3 Greater root = 2 Sum of Greater root of 3 equation = 5 + 5 + 2 = 12 Sum is 12
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