Given the point A(2,-4) and B(4, b) such that the distance between them is 12 units.

We will use the distance between two points formula to find the value of b.

==> We know that:

D = sqrt( x1-x2)^2 + (y1-y2)^2

==> sqrt( 4-2)^2 + (b+4)^2 = 12

==> sqrt( 4 + b^2 + 8b + 16) = 12

==> sqrt(b^2 + 8b + 20) = 12

Now we will square both sides.

==> b^2 + 8b + 20 = 144

==> b^2 + 8b + 20 - 144 = 0

==> b^2 + 8b -124 = 0

Now we will find the roots.

==> b1= ( -8 + sqrt( 64+496) /2

= ( -8 + 4sqrt35) /2 = ( -4+2sqrt35

==> b2= -4-2sqrt35

Then there are two possible values for b such that the distance between A and B is 12 .

**==> b = { -4+2sqrt35 , -4-2sqrt35 }**

The distance between two points (x1, y1) and (x2, y2) is given by sqrt ((x1 - x2)^2 + (y1 - y2)^2)

For the points A(2 , -4) and B(4, b) which are 12 units apart we have:

12 = sqrt ((2 - 4)^2 + (b + 4)^2)

=> 144 = 4 + (b + 4)^2

=> 144 = 4 + b^2 + 16 + 8b

=> 144 - 4 - 16 = b^2 + 8b

=> b^2 + 8b - 124 = 0

b1 = -8/2 + sqrt (64 + 496)/2

=> -4 + sqrt ( 560) / 2

b2 = -4 - (sqrt 560)/2

**The values that b can take are (-4 + sqrt ( 560) / 2 , -4 - (sqrt 560)/2**