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:

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

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.

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Step 1: Rewrite the Equation in Standard Form

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

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

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

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

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Step 2: Factor the Equation

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 $

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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.

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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.

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Summary

• Start with: $ 2t - t^2 = 0 $
• Rewrite and factor: $ t(t - 2) = 0 $
• Solutions: $ t = 0 $ and $ t = 2 $
• Select $ t = 2 $ as the valid non-zero solution

Mastering equations like this strengthens algebraic intuition and equips you to analyze more dynamic problems with confidence.

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Key Takeaway: In quadratic equations, decomposition into factors uncovers both trivial and essential solutions — always consider context to determine which solution truly matters.

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Use this method every time you encounter a quadratic — clarity comes from careful decomposition and thoughtful interpretation.