Question

Determine the values of `a` and `b`?

The equations `x^3+x^2+ax+b=0` and `x^3-bx^2-ax+4=0` both have `x=2` as a solution

Answer 1

`a=-10, b=8`

Explanation:

`x^3+x^2+ax+b=0`

has solution `x=2`

`=>2^3+2^2+2a+b=0`

`:.12+2a+b=0--(1)`

`x^3-bx^2-ax+4=0`

has solution `x=2`

`=>2^3-bxx2^2-2a+4=0`

`12-2a-4b=0--(2)`

`12+2a+b=0--(1)`

`(1)+(2)`

`24-3b=0`

`=>b=8`

sub into`(1)`

`12+2a+8=0`

`=>a=-10`

check in `(2)`

`12- 2xx-10 +8=12-20+8=0sqrt`

Answer 2

`a=-10" and "b=8`

Explanation:

`"since x = 2 is a solution to both equations we can"`
`"substitute this value directly into the equations"`

`rArr2^3+2^2+2a+b=0" and "`

`2^3-2^2b-2a+4=0`

`"simplifying to give"`

`8+4+2a+b=0rArr2a+b=-12to(1)`

`8-4b-2a+4=0rArr-2a-4b=-12to(2)`

`"adding equations "(1)" and "(2)" to eliminate a"`

`(2a-2a)+(b-4b)=-12+(-12)`

`rArr-3b=-24rArrb=8`

`"substitute this value in equation "(1)`

`2a+8=-12rArr2a=-20rArra=-10`

`rArra=-10" and "b=8`