As shown in the figure, we know that the image of quadratic function y = X2 - (M-3) x-m is a parabola. (1) when we try to find the value of M, the distance between the two intersections of parabola and x-axis is 3? (2) When m is the value, the two roots of the equation X2 - (M-3) x-m = 0 are negative? (3) Let m be the vertex of the parabola and P, Q be the intersection point of the parabola and x-axis

As shown in the figure, we know that the image of quadratic function y = X2 - (M-3) x-m is a parabola. (1) when we try to find the value of M, the distance between the two intersections of parabola and x-axis is 3? (2) When m is the value, the two roots of the equation X2 - (M-3) x-m = 0 are negative? (3) Let m be the vertex of the parabola and P, Q be the intersection point of the parabola and x-axis

(1) According to the meaning of the question, (M-3) 2-4 · (- M) 1 = 3, the solution is M1 = 0, M2 = 2, that is, when m is 0 or 2, the distance between the two intersections of the parabola and the X axis is 3; (2) ∵ = (M-3) 2-4 · (- M) = m2-2m + 9 = (m-1) 2 + 8 > 0, the equation X2 - (M-3) x-m = 0 has two real roots, let the two roots of the equation X2 - (M-3) x-m = 0 be x1, X2, then X1 + x2 = M-3 < 0, x1 ·x2 = - M > 0, ∵ m < 0; (3) When ∵ PQ = (M-3) 2-4 · (- M) = (m-1) 2 + 8, ∵ M = 1, PQ is the shortest, and the shortest value is 8 = 22. At this time, the analytical formula of parabola is y = x2 + 2x-1 = (x + 1) 2-2, ∵ m point coordinates are (- 1, - 2), ∵ MPQ area = 12 × 2 × 22 = 22