solve (12/t)+t-8=0 (solving rational equations)
Carli, I do not recall seeing algebraic equations in a calculus course.
(12/t)+t-8=0
Assuming t≠0, multiply by t to get:
t(12/t)+t²-8t=0
t²-8t+12 = 0
Factor and solve for t.
To solve the rational equation (12/t) + t - 8 = 0, follow these steps:
Step 1: Clear the fraction by multiplying all terms by the common denominator, which is t. This will help eliminate the fraction.
t * (12/t) + t * t - 8 * t = 0
Simplifying,
12 + t^2 - 8t = 0
Step 2: Rearrange the equation to bring all terms to one side, so the equation becomes a quadratic equation.
t^2 - 8t + 12 = 0
Step 3: Factor the quadratic equation. Look for two numbers that multiply to give you 12 and sum to give you -8.
(t - 6)(t - 2) = 0
Step 4: Set each factor to zero and solve for t.
t - 6 = 0 or t - 2 = 0
For t - 6 = 0, add 6 to both sides:
t = 6
For t - 2 = 0, add 2 to both sides:
t = 2
Step 5: Check the solution(s) by substituting the values back into the original equation.
For t = 6:
(12/6) + 6 - 8 = 0
2 + 6 - 8 = 0
0 = 0 (True)
For t = 2:
(12/2) + 2 - 8 = 0
6 + 2 - 8 = 0
0 = 0 (True)
Therefore, the solutions to the rational equation (12/t) + t - 8 = 0 are t = 6 and t = 2.