Question

If both x – 2 and x -(1/2)  are factors of px² + 5x + r, then show that p = r.

Answer 1

Given, f(x) = px²+5x+r and factors are x-2, x – ½

g₁(x) = 0,

x – 2 = 0

x = 2

Substituting x = 2 in place of equation, we get

f(x) = px² + 5x + r

f(2) = p(2)² + 5(2) + r = 0

=  4p + 10 + r = 0   … eq.(i)

x – ½ = 0

x = ½

Substituting x = ½ in place of equation, we get,

f(x) = px² + 5x + r

f( ½ ) = p( ½ )² + 5( ½ ) + r =0

= p/4 + 5/2 + r = 0

= p + 10 + 4r = 0  … eq(ii)

On solving eq(i) and eq(ii),

We get,

4p + r = – 10 and p + 4r = – 10

Since the RHS of both the equations are same,

We get,

4p + r = p + 4r

3p=3r

p = r.

Hence Proved.