Question

The expression `10x^2-x-24` can be written as `(Ax-8)(Bx+3)`, where `A` and `B` are integers. What is `AB + B`?

Question from AoPS. Please help :)

Answer 1

`AB+B=12, 65/8`

Explanation:

.

`10x^2-x-24=(Ax-8)(Bx+3)`

`10x^2-x-24=ABx^2+3Ax-8Bx-24`

`10x^2-x-24=ABx^2-(8B-3A)x-24`

`AB=10`

`8B-3A=1, :. 8B=1+3A, B=(1+3A)/8`

`(A(1+3A))/8=10`

`3A^2+A-80=0`

`A=(-1+-sqrt(1-4(3)(-80)))/6=(-1+-sqrt961)/6=(-1+-31)/6`

`A=5, -16/3`

`A=5, :. B=2, :. AB+B=10+2=12`

`A=-16/3, :. B=-15/8, :. AB+B=10-15/8=65/8`