Question

If both (x-3) and (x -1/3) are factors of ax^2 + 5x + b , then show that a = b

Answer 1

Given :-

p ( x ) = ax² + 5x + b

( x - 3 ) & ( x - ⅓ ) are the factors of p ( x )

Required to find :-

• a = b

Method used :-

• Factor theorem

Solution :-

Given :-

p ( x ) = ax² + 5x + b

Since,

( x - 3 ) is a factor .

So,

Let ,

x - 3 = 0

x = 3

p ( 3 ) =

a ( 3 )² + 5 ( 3 ) + b = 0

a ( 9 ) + 15 + b = 0

9a + 15 + b = 0

b = - 9a - 15 $\longrightarrow{\text{Equation - 1 }}$

consider this as equation 1

However

It is also mentioned that ;

( x - ⅓ ) is also an factor of p ( x )

So,

Let,

x - ⅓ = 0

x = ⅓

p ( x ) = ax² + 5x + b

p ( ⅓ ) =

a ( 1/3 )² + 5 ( 1/3 ) + b = 0

a ( 1/9 ) + 5/3 + b = 0

a/9 + 5/3 + b = 0

substitute the value of b from equation 1

So,

a/9 + 5/3 + ( - 9a ) + ( - 15 ) = 0

a/9 + 5/3 - 9a - 15 = 0

a + 15 - 81a - 135 / 9 = 0

- 80a - 120 / 9 = 0

- 80a - 120 = 0 x 9

- 80a - 120 = 0

- 80a = 120

a = 120/-80

a = 12/-8

a = - 3/2

Substitute the value of a in equation 1

So,

b = - 9a - 15

b = - 9 ( - 3/2 ) - 15

b = 27 / 2 - 15

b = 27 - 30 / 2

b = - 3/2

$\large{\dagger{\boxed{\rm{\therefore{ a = b = \dfrac{-3}{2}}}}}}$

✅ Hence Solved