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.

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