This is Q25 form Cambridge Mathematics year 11 3 unit (in case anyone has the book rather than following me)

(a) show that the tangent to P:y=ax^2+bx+c with gradient m has y-intercept c-[(m-b)^2]/4a.

y'=2ax+b, which is m

rearrange for x=m-b/2a

sub x into y gives y=a(m-b/2a)^a+b(m-b/2a)+c

then I used the gradient/point formula and the above values for m, x and y which simplifed to give y-intercept c-[(m-b)^2]/4a.

So part a is done

(b) Hence find any equations of any quadratics that pass through the origin and are tangents to both y=-2x-4 and y=8x-49.

y=-2x-4, y'=-2x therefore m=-2 and c=0 (1)

y=8x-49, y'=8 therefore m=8 and c=0 (2)

I sub'ed the values from (1) and (2) into y-intercept=c-[(m-b)^2]/4a and got

16a=b^2+4b+4 (1)

196a=b^2-16b-64 (2)

and then solved simultaneous equations and found b=0 or 6

Now sub b=6 back onto (1) or (2) gives a=1 which gives

y=x^2-6x and that is one of the answers

the second answer is (25/81)x^2+(2/9)x

I cannot find this second answer. Maybe there is something I have missed. Can someone please help me.

Thank you.