Question

If x + a is a common factor of expressions f(x) = x² + px + q and g(x) = x² + mx + n; show that : `a = (n - q)/(m - p)` 

Answer 1

f(x) = x² + px + q

It is given that (x + a) is a factor of f(x).

∴ f(–a) = 0 

`\implies` (–a)² + p(–a) + q = 0 

`\implies` a² – pa + q = 0

`\implies` a² = pa – q   ...(i) 

g(x) = x² + mx + n

It is given that (x + a) is a factor of g(x). 

∴ g(–a) = 0

`\implies` (–a)² + m(–a) + n = 0

`\implies` a² – ma + n = 0

`\implies` a² = ma – n   ...(ii) 

From (i) and (ii), we get, 

pa – q = ma – n

n – q = a(m – p)

`a = (n - q)/(m - p)` 

Hence, proved.