Question

The expression $12y^2-65y+42$ can be written as $(Ay-14)(By-3),$ where $A$ and $B$ are integers. What is $AB + A$?

Answer 1

`AB + A = 16`

Explanation:

Given

`12y^2 - 65y + 42 = (Ay -14)(By - 3)`

Required

Find
`AB + A`

`12y^2 - 65y + 42 = (Ay -14)(By - 3)`

Expand

`12y^2 -56y -9y + 42 = (Ay - 14)(By -3)`

Factorize

`4y(3y -14) -3(3y - 14) = (Ay - 14)(By -3)`

Factor out 3y- 14

`(3y - 14)(4y -3) = (Ay - 14)(By -3)`

By comparison:

`Ay - 14 = 3y - 14`

and

`By - 3 = 4y - 3`

So, we have:

`Ay - 14 = 3y - 14`

Add 14 to both sides

`Ay = 3y`

Divide both sides by y

`A = 3`

`By - 3 = 4y - 3`

Add 3 to both sides

`By = 4y`

Divide both sides by y

`B = 4`

So, the expression
`AB + A` is:

`AB + A = 3 * 4 + 4`

`AB + A = 12 + 4`

`AB + A = 16`