Question 295465
Solving for the specified variable
x = vt -1/2at^2; t
Assume the formula
x = {{{vt - (1/2)a*t^2)}}}
Multiply by 2 to get rid of the denominator
{{{2x = 2vt - a*t^2}}}
arrange this as a quadratic equation on the left
at^2 - 2vt + 2x = 0
Use the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In our problem: x=t; a=a; b=-2v, c=2x
{{{t = (-(-2v) +- sqrt((-2v)^2- 4*a*2x ))/(2*a) }}}
:
{{{t = (2v +- sqrt(4v^2 - 8ax ))/(2*a) }}}
Factor out 4 inside the radical
{{{t = (2v +- sqrt(4(v^2 - 2ax) ))/(2*a) }}}
extract the square root of 4
{{{t = (2v +- 2*sqrt(v^2 - 2ax ))/(2*a) }}}
cancel out the 2's, we have two solutions
{{{t = (v + sqrt(v^2 - 2ax ))/(a) }}}
and
{{{t = (v - sqrt(v^2 - 2ax ))/(a) }}}