Question

solve for t. if there are multiple solutions, enter them as a list of values separated by commas. if there are no solutions, enter none.
\\(\dfrac{12}{t} + \dfrac{18}{t + 11} = \dfrac{9}{2}\\)

Explanation:

Step1: Find common denominator

The common denominator of \(t\), \(t + 11\), and \(2\) is \(2t(t + 11)\). Multiply each term by \(2t(t + 11)\) to eliminate the fractions:

$$ \frac{12}{t}\times2t(t + 11)+\frac{18}{t + 11}\times2t(t + 11)=\frac{9}{2}\times2t(t + 11) $$

Simplify each term:
\(12\times2(t + 11)+18\times2t = 9t(t + 11)\)
\(24(t + 11)+36t = 9t^{2}+99t\)

Step2: Expand and simplify

Expand \(24(t + 11)\):
\(24t+264 + 36t=9t^{2}+99t\)
Combine like terms on the left:
\(60t + 264=9t^{2}+99t\)
Rearrange to form a quadratic equation:
\(9t^{2}+99t - 60t - 264 = 0\)
\(9t^{2}+39t - 264 = 0\)
Divide all terms by \(3\) to simplify:
\(3t^{2}+13t - 88 = 0\)

Step3: Solve the quadratic equation

Use the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\). Here, \(a = 3\), \(b = 13\), \(c=-88\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(13)^{2}-4\times3\times(-88)=169 + 1056 = 1225\)
Then, \(t=\frac{-13\pm\sqrt{1225}}{2\times3}=\frac{-13\pm35}{6}\)

Step4: Find the two solutions

For the plus sign:
\(t=\frac{-13 + 35}{6}=\frac{22}{6}=\frac{11}{3}\)
For the minus sign:
\(t=\frac{-13 - 35}{6}=\frac{-48}{6}=-8\)

Step5: Check for extraneous solutions

We need to check if these solutions make the original denominators zero.
For \(t=\frac{11}{3}\), \(t
eq0\) and \(t + 11=\frac{11}{3}+11=\frac{11 + 33}{3}=\frac{44}{3}
eq0\), so it's valid.
For \(t=-8\), \(t
eq0\) and \(t + 11=-8 + 11 = 3
eq0\), so it's valid.

Answer:

\(-8,\frac{11}{3}\)