F.4 Quadratic equation CRAZY!!

Let m and n be the roots of ax2 + bx + c = 0

m and s be the roots of px2 + qx + r = 0

Such that the equations have the common root m.

Sum of roots, m + n = -b/a ... (1)

m + s = -q/p ... (2)

Product of roots, mn = c/a ... (3)

ms = r/p ... (4)

So, From (2), s = -q/p - m ... (5)

From (1), n = -b/a - m ... (6)

(3)/(4): n/s = cp / ar

(-b/a - m) / (-q/p - m) = cp / ar

br + mar = cq + mcp

m = (cq - br) / (ar - cp) ... (7)

Put (7) into (5): s = -q/p - m = -q/p - (cq - br)/(ar - cp)

Put these results into (4):

[(cq - br) / (ar - cp)][-q/p - (cq - br)/(ar - cp)] = r/p

-q(cq - br)/p(ar - cp) - (cq - br)2/(ar - cp)2 = r/p

-q(ar - cp)(cq - br) - p(cq - br)2 = r(ar - cp)2

(br - cq)[qar - cpq + pcq - pbr] = r(cp - ar)2

(br - cq)(aq - bp) = (cp - ar)2






2009-10-10 14:53:27 補充：
It should be (br - cq)(aq - bp) = (cp - ar)^2

(br - cq)[qar - cpq + pcq - pbr] = r(cp - ar)2

(br - cq)(aq - bp) = (cp - ar)2

這步何來？？