If both (x – 2) and (x – 1/2) are the factors of ax^2 + 5x + b, then show that a = b.

Question

If both (x – 2) and (x – \(\frac{1}{2}\)) are the factors of ax² + 5x + b, then show that a = b.

Answer 1

Let P(x) = ax² + 5x + b 

(x – 2) is a factor of P(x), if P(2) = 0 

P(2) = a(2)² + 5(2) + b = 0

4a + 10 + b = 0 

4 a + b = – 10 … (1)

(x – \(\frac{1}{2}\)) is a factor of P(x), P(\(\frac{1}{2}\)) = 0

2a + 8b = -20 

a + 4b = – 10 … (2) 

From (1) and (2) 

4a + b = – 10 … (1) 

a + 4b = – 10 … (2) 

(1) and (2) ⇒ 4a + b = a + 4b 

3a = 3b 

∴ a = b. Hence it is proved.