Question

Solve for t(advanced rational equations)

Answer 1

t² -7 + 12t⁻² = 0

Multiply each side by t² :

t⁴ -7t² + 12 = 0

This is a quadratic equation in the variable ' t² '.
For just a moment, to avoid confusion, let U = t².
Then the equation is

U² - 7U + 12 = 0
whence
(U - 4) (U - 3) = 0
and
U = 4   and   U = 3 .

Now we can go back to ' t² ' in place of U :

t² = 4
t = +2  and  t = -2

t² = 3
t = +√3  and  t = -√3

                                       

Answer 2

$t^2-7+12t^{-2}=0\\ t^4-7t^2+12=0\\ t^4-3t^2-4t^2+12=0\\ t^2(t^2-3)-4(t^2-3)=0\\ (t^2-4)(t^2-3)=0\\ (t-2)(t+2)(t^2-3)=0\\ t=2 \vee t=-2 \vee t=-\sqrt{3}\vee t=\sqrt3$