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
t 2 − 7 + 12 t − 2 = 0 t 4 − 7 t 2 + 12 = 0 t 4 − 3 t 2 − 4 t 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 ∨ t = − 2 ∨ t = − 3 ∨ t = 3
To solve the advanced rational equation, we can use the quadratic formula, which provides solutions based on the coefficients of the equation. The discriminant can indicate if the roots are real or complex. For instance, the equation 3 t 2 + t − 4 = 0 yields two real solutions: t = 1 and t = − 3 4 , while 2 t 2 + 6 t + 5 = 0 has no real solutions but two complex solutions.
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