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📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 Question:** A science educator is designing a lesson on quadratic equations and presents the equation \( x^2 - 5x + 6 = 0 \). They ask students to compute the product of the roots.

📌 Solution:** For the quadratic equation \( x^2 - 5x + 6 = 0 \), Vieta's formulas tell us that the product of the roots is given by \( \frac{d}{a} \), where \( a \) is the coefficient of \( x^2 \) and \( d \) is the constant term. Here, \( a = 1 \) and \( d = 6 \), so the product of the roots is:

📌 \frac{d}{a} = \frac{6}{1} = 6

📌 The product of the roots is \(\boxed{6}\).

📌 Question:** A palynologist is modeling a real-world phenomenon with the cubic polynomial \( g(x) = 2x^3 - 3x^2 + 5x - 7 \). Find \( g(2) \).

📌 Solution:** To evaluate \( g(2) \) for the polynomial \( g(x) = 2x^3 - 3x^2 + 5x - 7 \), substitute \( x = 2 \) into the polynomial:

📌 g(2) = 2(2)^3 - 3(2)^2 + 5(2) - 7 = 2 \cdot 8 - 3 \cdot 4 + 10 - 7

📌 = 16 - 12 + 10 - 7 = 16 - 12 = 4; \quad 4 + 10 = 14; \quad 14 - 7 = 7

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