quadratic equations in one unknow

(a)
y=m+nx-x2
Since the x-axis at two points P(-1/2,0) and Q(1,0),
y=0 when x=-1/2 or 1
m+nx-x2=0 when x=-1/2 or 1
Therefore the solution is x=-1/2 or 1.

(b)
Substitute x=-1/2 into the quadratic equation m+nx-x2=0,
m+nx-x2=0
m+n(-1/2)-(-1/2)2=0
m-n/2-1/4=0
4m-2n-1=0 (i)
Substitute x=1 into the quadratic equation m+nx-x2=0,
m+n(1)-(1)2=0
m+n-1=0 (ii)
(ii)×2
2m+2n-2=0 (iii)
(i)+(iii)
6m-3=0
6m=3
m=1/2
Substitute m=1/2 into (ii),
m+n-1=0
1/2+n-1=0
n=1/2

a)from the graph when y=0,x=-1/2 or x=1
b)
(-1/2)+1=n...........(by sum of roots)
.......n=1/2
(-1/2)(1)=-m..........(product of roots)
......m=1/2

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