Solve: 27/t-7=t+7/3
The way you typed it :
27/t - 7 = t + 7/3
multiply each term by 3t
81 - 21t = t^2 + 7t
t^2 + 28t - 81 = 0
t^2 + 28t + 196 = 81+196
(t+14)^2 = 277
t+14 = ± √277
t = -14± √277
but...
I have a feeling you mean
27/(t-7) = (t+7)/3
cross-multiply
t^2 - 49 = 81
t^2 = 130
t = ± √130
in my first solution, the third line should have been
81 - 21t = 3t^2 + 7t
3t^2 + 28t -81 = 0
t = (-28 ± √1756)/6
= (-28 ± 2√439)/6
= (-14 ± √439)/3
To solve the equation 27/t - 7 = t + 7/3, we can follow these steps:
Step 1: Multiply both sides of the equation by the common denominator, which is 3t. This will eliminate the denominators in the equation.
3t * (27/t) - 3t * 7 = 3t * (t + 7/3)
Step 2: Simplify each term.
81 - 21t = 3t^2 + 7t
Step 3: Rearrange the equation in standard form by moving all the terms to one side of the equation.
3t^2 + 7t + 21t - 81 = 0
Step 4: Combine like terms.
3t^2 + 28t - 81 = 0
Step 5: Solve the quadratic equation. You can either factor it, use the quadratic formula, or complete the square. Factoring is not possible in this case, so we will use the quadratic formula.
The quadratic formula is given by:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, the coefficients are:
a = 3, b = 28, c = -81
Plugging those values into the formula, we get:
t = (-28 ± sqrt(28^2 - 4 * 3 * -81)) / (2 * 3)
Step 6: Simplify the equation.
t = (-28 ± sqrt(784 + 972)) / 6
= (-28 ± sqrt(1756)) / 6
= (-28 ± sqrt(4 * 439)) / 6
= (-28 ± 2sqrt(439)) / 6
Step 7: Further simplify the equation.
t = (-14 ± sqrt(439)) / 3
So the solutions to the equation 27/t - 7 = t + 7/3 are:
t = (-14 + sqrt(439)) / 3
t = (-14 - sqrt(439)) / 3