Final answer:
To solve the equation t+4/t+3/t-4=-16/t^2-4t, first find a common denominator. Simplify both sides of the equation, combine like terms, and rearrange it into a quadratic equation. Solve the quadratic equation using the quadratic formula.
Step-by-step explanation:
To solve the equation t+4/t+3/t-4=-16/t^2-4t, we can first simplify the left side of the equation by finding a common denominator. Multiply the first fraction t+4 by (t-4) and the second fraction t-3 by t, so that both fractions have the same denominator.
This gives us (t(t-4)+4(t-4))/(t(t-4)) + (t(t-3))/(t(t-4)) = -16/(t^2-4t).
Simplifying further, we get (t^2-4t+4t-16)/(t(t-4)) + (t^2-3t)/(t(t-4)) = -16/(t^2-4t).
Combine like terms to obtain (t^2-3t)/(t(t-4)) = -16/(t^2-4t).
Now, multiply both sides of the equation by t(t-4) to eliminate the denominators. This gives us t^2 - 3t = -16.
Next, rearrange the equation to get it in the form of a quadratic equation: t^2 - 3t + 16 = 0.
Finally, we can solve the quadratic equation using the quadratic formula: t = (-b ± √(b^2 - 4ac))/(2a).
Substituting the values a=1, b=-3, and c=16 into the quadratic formula, we get t = (3 ± √((-3)^2 - 4(1)(16)))/(2(1)).
Simplifying further, we have t = (3 ± √(9 - 64))/2.
This results in t = (3 ± √(-55))/2.
Since the square root of a negative number is not a real number, the solutions for t in this equation are complex numbers.