y'=10x+a=0 at a turning point, so x=-a/10. The point is a minimum (because y">0). When x=-a/10, y=5a^2/100-a^2/10+b=-a^2/20+b. The turning point is at (-a/10,b-a^2/20) but as this is (b,a), b=-a/10 and a=b-a^2/20=-a/10-a^2/20. So 20a=-2a-a^2 or 22a=-a^2. Since a not zero, a=-22 and b=-a/10=2.2. The equation becomes y=5x^2-22x+2.2, with a=-22 and b=2.2.