Question

If m,n are roots of equation x^{​​​​​​2} +px +q and g,h are roots of equation x^{​​​​​​2} + px - r . Find (m-g)(m-g).

Answer 1

The information given is,

x^2 + px - r = 0 has roots g.h
x^2 + px + q = 0 has roots m,n

Using Sum of roots and Product of roots formulae on both these equations, we get these results-

g + h = -p (1)
gh = -r (2)

m + n = -p (3)
mn = q (4)



We will begin with (α - γ)(α - δ).


=> (m - g)(m - h)
=> m^2 - mh - mg + gh
=> m^2 - m(h + g) + gh

From (3) and (1), it is understood that (m + n) = (g + h). We'll use it here,

=> m^2 - m(m + n) + gh
=> m^2 - m^2 - mn + gh
=> - mn + gh

From (2) and (4), we know that mn = -q and gh = r. Using these here, we'll get

=> -(-q) + r
=> q + r