Answer: -4.
- A -4
- B -2
- C 0
- D 4
Correct answer: A. -4
Explanation: For x²+px+q=0 with roots differing by 1: (α-β)²=(α+β)²-4αβ = p²-4q = 1. Similarly for x²+qx+p=0: q²-4p = 1. Subtract: p²-q² = 4p-4q → (p-q)(p+q) = 4(p-q). If p≠q, then p+q = 4. Add the two equations: p²+q²-4(p+q)=2. With p+q=4: p²+q²=18. But (p+q)²=p²+2pq+q²=16, so 2pq=-2, pq=-1. Then p and q are roots of t²-4t-1=0, real but not equal. Alternatively factoring: (p-q)(p+q+4)=0 gives p+q=-4 when using the subtraction the other way. Re-doing: p²-4q=1 and q²-4p=1. Subtract: p²-q²-4q+4p=0 → (p-q)(p+q)+4(p-q)=0 → (p-q)(p+q+4)=0. So p+q=-4 (if p≠q).
A complex number z = a+ib is plotted as a point (or vector from the origin) on the Argand plane, with the real part along the horizontal axis and the imaginary part along the vertical; its modulus |z| is the vector's length, and its argument θ is the angle from the positive real axis.
Concept context
Complex numbers in a+ib form, modulus, argument, polar form, cube roots of unity, and quadratic equations with complex roots
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