Let's solve the equation step-by-step:

\[ -7 = \frac{(t - 11)(11 - t)}{3} \]

First, simplify the expression \((t - 11)(11 - t)\).

Notice that:

\[ (t - 11)(11 - t) = (t - 11)(- (t - 11)) = - (t - 11)^2 \]

So, we have:

\[ -7 = \frac{- (t - 11)^2}{3} \]

Next, multiply both sides by 3 to eliminate the fraction:

\[ -21 = - (t - 11)^2 \]

Since the negatives on both sides cancel each other out, we get:

\[ 21 = (t - 11)^2 \]

To solve for \(t\), take the square root of both sides:

\[ \sqrt{21} = \pm (t - 11) \]

This gives us two solutions:

\[ t - 11 = \sqrt{21} \]
\[ t - 11 = -\sqrt{21} \]

Solving these equations for \(t\):

1. \( t = 11 + \sqrt{21} \)
2. \( t = 11 - \sqrt{21} \)

So, the solutions are:

\[ t = 11 + \sqrt{21} \]
\[ t = 11 - \sqrt{21} \]

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you simplify expressions involving products of binomials?
2. What are the properties of square roots that can help solve quadratic equations?
3. How do you handle equations with fractions?
4. What are other methods to solve quadratic equations?
5. How do you interpret and graph solutions of quadratic equations?

**Tip:** When solving quadratic equations, always check if the quadratic expression can be factored for a simpler solution.

Learn how to solve the quadratic equation -7 = (t-11)(11-t)/3 with a step-by-step solution. This problem involves simplifying expressions, solving equations with fractions, and applying the square root method. Suitable for students in Grades 8-10.

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