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why is this the case instead of LHS equals ((t-1)^2)/(1+t^2)

Let's explore why the original equation is:

\[
\frac{1 + t^2 - 2t}{1 + t^2} = \frac{(1 - t)^2}{1 + t^2}
\]

and not:

\[
\frac{1 + t^2 - 2t}{1 + t^2} = \frac{(t - 1)^2}{1 + t^2}.
\]

### Step 1: Expand and Compare the Right-Hand Sides
We already know from the earlier explanation that:

\[
(1 - t)^2 = 1 - 2t + t^2.
\]

Now, let’s expand \((t - 1)^2\):

\[
(t - 1)^2 = t^2 - 2t + 1.
\]

### Step 2: Analyze the Expressions
Notice that:

\[
1 - 2t + t^2 \quad \text{(from } (1 - t)^2\text{)}
\]

and

\[
t^2 - 2t + 1 \quad \text{(from } (t - 1)^2\text{)}
\]

are identical. In fact, these expressions are equivalent:

\[
1 - 2t + t^2 = t^2 - 2t + 1.
\]

### Step 3: Consider the Form
The expression \(\frac{(1 - t)^2}{1 + t^2}\) is typically preferred over \(\frac{(t - 1)^2}{1 + t^2}\) due to convention. The standard form for writing polynomials usually starts with the constant term and follows with the terms in decreasing powers of \(t\).

In this case, although both \((1 - t)^2\) and \((t - 1)^2\) expand to the same expression \(1 - 2t + t^2\), the choice to use \((1 - t)^2\) maintains the order starting with the constant, then the linear term, and finally the quadratic term:

\[
1 - 2t + t^2.
\]

Mathematically, both are correct, but the expression \((1 - t)^2\) is used for clarity and consistency, keeping the terms in a more standard order.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How do different factorizations of a quadratic expression influence its graph?
2. Can you think of a scenario where choosing \((t-1)^2\) might be more advantageous?
3. How do transformations affect the form of algebraic expressions?
4. What are the implications of different orders in polynomial terms in solving equations?
5. How does expanding a square like \((1-t)^2\) relate to completing the square in solving quadratic equations?

**Tip:** When simplifying or factoring expressions, consider the standard conventions in algebra. These conventions can make the expressions easier to interpret and compare.

Explore why the expression (1 - t)^2 = (t - 1)^2 in algebraic equations and the implications of choosing between these forms. Learn through detailed steps and comparisons. Suitable for students in Grades 10-12.

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