Question

Two factors of 24 add up to 11. What are they?​

Answer 1

The numbers are
`8` and
`3`.

Explanation:

Let
`a` and
`b` the numbers.

If
`a` and
`b` are factors of
`24`, then
`ab = 24`

If
`a` and
`b` has to add up to
`11`, the
`a +b = 11`

Solving for
`a` in terms of
`b` in the equation,
`a +b = 11`:

`a +b = 11 \\ a +b -b = 11 -b \\ a = 11 -b`

Solving the equation
`ab = 24` by plugging in
`a`:

`ab = 24 \\ (11 -b)b = 24 \\ -b^2 +11b = 24 \\ b^2 -11b = -24 \\ b^2 -11b +(121)/(4) = -24 +(121)/(4) \\ (b -(11)/(2))^2 = -(96)/(4) +(121)/(4) \\ (b -(11)/(2))^2 = (25)/(4) \\ \sqrt{(b -(11)/(2))^2} = \pm \sqrt{(25)/(4)} \\ b -(11)/(2)= \pm (5)/(2)`

Solving
`b` from the positive root:

`b -(11)/(2) +(11)/(2) = (5)/(2) +(11)/(2) \\ b = (16)/(2) \\ b = 8`

Solving
`b` from the negative root:

`b -(11)/(2) +(11)/(2) = -(5)/(2) +(11)/(2) \\ b = (6)/(2) \\ b = 3`

Solving for
`a` in the equation,
`a = 11 -b` when
`b = 3`:

`a = 11 -(3) \\ a = 11 -3 \\ a = 8`

Solving for
`a` in the equation
`a = 11 -b` when
`b = 8`:

`a = 11 -(8) \\ a = 11 -8 \\ a = 3`

Answer 2

My answer is in the photo above