Understanding Option B: Solving the Quadratic Equation $2t - t^2 = 0$

When faced with a quadratic equation, breaking it down step-by-step not only reveals the full solution set but also deepens your understanding of its structure. One elegant example is the equation:

$$
2t - t^2 = 0
$$

Understanding the Context

This equation appears simple but offers a clear path through factoring, yielding a meaningful and valid non-zero solution: $ t = 2 $. Let’s explore how to solve it and interpret its significance.

Step 1: Rewrite the Equation in Standard Form

To solve $ 2t - t^2 = 0 $, arrange the terms in standard quadratic form:

Option B: $2t - t^2 = 0$ → $t(2 - t) = 0$, so $t = 2$ is valid non-zero solution.

Key Insights

$$
-t^2 + 2t = 0
$$

For easier factoring, multiply both sides by $-1$:

$$
t^2 - 2t = 0
$$

Step 2: Factor the Equation

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Final Thoughts

Now, factor out the common term $t$:

$$
t(t - 2) = 0
$$

This product equals zero if and only if one of the factors is zero. Using the zero product property:

$ t = 0 $
$ t - 2 = 0 $ → $ t = 2 $

Step 3: Identify Valid, Non-Zero Solutions

While $ t = 0 $ is a valid mathematical solution, $ t = 2 $ stands out as the non-zero, meaningful solution. In many real-world applications—such as modeling profit, motion, or optimization—$ t = 0 $ often represents a trivial or invalid state, making $ t = 2 $ the meaningful choice.

Why This Matters: Practical Insight

This equation might model scenarios like revenue derived from pricing: suppose a quadratic function models total revenue based on price $ t $, and $ t = 0 $ yields no revenue, while $ t = 2 $ represents the optimal price yielding maximum output. Recognizing $ t = 2 $ as the non-zero solution helps focus on behavior beyond trivial points.