Option B: $2t - t^2 = 0$ → $t(2 - t) = 0$, so $t = 2$ is valid non-zero solution. - Redraw
Understanding Option B: Solving the Quadratic Equation $2t - t^2 = 0$
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:
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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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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.
