Question

How do I solve the formula `16t^2 - vt - 40 = 0` for `t`?

Answer 1

`t = (v +-sqrt(v^2+2560))/32`

Explanation:

We have:

`16t^2-vt-40 = 0`

We can complete the square, by first dividing by the coefficient of `t^2`

`:. 16{t^2-v/16t-40/16} = 0`

`:. t^2-v/16t-5/2 = 0`

Then we factor half the coefficient of `t`:

`:. (t-v/32)^2-(v/32)^2-5/2 = 0`

Which we can now readily rearrange for `t`:

`:. (t-v/32)^2 = v^2/1024+5/2`

`:. (t-v/32)^2 = v^2/1024+2560/1024`

`:. (t-v/32)^2 = (v^2+2560)/1024`

`:. t-v/32 = +-sqrt((v^2+2560)/1024)`

`:. t-v/32 = +-sqrt(v^2+2560)/32`

`:. t = v/32 +-sqrt(v^2+2560)/32`

`:. t = (v +-sqrt(v^2+2560))/32`