Question

If `x = 2/3` and x = –3 are the roots of the equation ax² + 7x + b = 0, find the values of a and b.

Answer 1

We have been given that,

`ax^2 + 7x + b = 0, x = 2/3, x = -3`

We have to find a and b

Now, if `x = 2/3` is a root of the equation, then it should satisfy the equation completely. Therefore we substitute `x = 2/3` in the above equation. We get,

`a(2/3)^2 + 7(2/3) + b = 0`

`(4a + 42+9b)/9=0`

`a=(-9b-42)/4`                  .........(1)

Also, if x = -3 is a root of the equation, then it should satisfy the equation completely. Therefore we substitute x = -3 in the above equation. We get,

a(-3)² + 7(-3) + b = 0

9a - 21 + b = 0                 .........(2)

Now, we multiply equation (2) by 9 and then subtract equation (1) from it. So we have,

81a + 9b - 189 - 4a - 9b - 42 = 0

77a - 231 = 0

`a = 231/77`

a = 3

Now, put this value of ‘a’ in equation (2) in order to get the value of ‘b’. So,

9(3) + b - 21 = 0

27 + b - 21 = 0

b = 21 - 27

b = -6

Therefore, we have a = 3 and b = -6.