Question

Find b and then solve the equation: d (b−5)x2−(b−2)x+b=0, if one of its roots is 1/2

Answer 1

`b = (1)/(3)`

`x = (1)/(2) \: or \: x = - (1)/(7)`

Step-by-step explanation

The given expression is

`(b - 5) {x}^(2) - (b - 2)x + b = 0`

If

`x = (1)/(2)`

is a root, then it must satisfy the given equation.

`(b - 5) {( (1)/(2) )}^(2) - (b - 2)( (1)/(2) )+ b = 0`

`(b - 5) {( (1)/(4) )} - (b - 2)( (1)/(2) )+ b = 0`

Multiply through by 4,

`(b - 5)- 2(b - 2)+4 b = 0`

Expand:

`b - 5- 2b + 4+4 b = 0`

Group similar terms;

`b - 2b + 4b = 5 - 4`

`3b = 1`

`b = (1)/(3)`

Our equation then becomes:

`( (1)/(3) - 5) {x}^(2) - ( (1)/(3) - 2)x + (1)/(3) = 0`

`( - (14)/(3) ) {x}^(2) - ( - (5)/(3) )x + (1)/(3) = 0`

`- 14{x}^(2) + 5x + 1= 0`

Factor:

`(2x - 1)(7x + 1) = 0`

`x = (1)/(2) \: or \: x = - (1)/(7)`