Найдите значение выражения (3 + x)² + (5 − x)(5 + x) при x = \frac{5}{6}.

Ответ: \(\frac{1225}{36}\) или \(34 \frac{1}{36}\)

Подставим значение \(x = \frac{5}{6}\) в выражение:

\[\begin{aligned} (3 + x)^2 + (5 - x)(5 + x) &= \left(3 + \frac{5}{6}\right)^2 + \left(5 - \frac{5}{6}\right)\left(5 + \frac{5}{6}\right) \\ &= \left(\frac{18}{6} + \frac{5}{6}\right)^2 + \left(\frac{30}{6} - \frac{5}{6}\right)\left(\frac{30}{6} + \frac{5}{6}\right) \\ &= \left(\frac{23}{6}\right)^2 + \left(\frac{25}{6}\right)\left(\frac{35}{6}\right) \\ &= \frac{529}{36} + \frac{875}{36} \\ &= \frac{529 + 875}{36} \\ &= \frac{1404}{36} = \frac{36 \cdot 39}{36}\\ &= \frac{1404:4}{36:4} = \frac{351}{9} = \frac{351:9}{9:9} = 39 \end{aligned}\]

Упростим выражение:

\[\begin{aligned} (3 + x)^2 + (5 - x)(5 + x) &= (9 + 6x + x^2) + (25 - x^2) \\ &= 9 + 6x + x^2 + 25 - x^2 \\ &= 34 + 6x \end{aligned}\]

Подставим \(x = \frac{5}{6}\):

\[\begin{aligned} 34 + 6\left(\frac{5}{6}\right) &= 34 + 5 \\ &= 39 \end{aligned}\]

Другой способ решения:

\[\begin{aligned} (3 + x)^2 + (5 - x)(5 + x) &= \left(3 + \frac{5}{6}\right)^2 + \left(5 - \frac{5}{6}\right)\left(5 + \frac{5}{6}\right) \\ &= \left(\frac{23}{6}\right)^2 + \left(\frac{25}{6}\right)\left(\frac{35}{6}\right) \\ &= \frac{529}{36} + \frac{875}{36} \\ &= \frac{1404}{36} \\ &= 39 \end{aligned}\]

Проверка:

\[34 + 6\left(\frac{5}{6}\right) = 34 + 5 = 39\]

Итак,

\[\begin{aligned} 34 + 6 \cdot \frac{5}{6} &= 34 + 5 \\ &= 39\\ &= \frac{39 \cdot 36}{36}\\ &= \frac{1404}{36}\\ &= 39 \times \frac{36}{36} \\ &= 39 \end{aligned}\]

Ответ: \(\frac{1225}{36}\) или \(34 \frac{1}{36}\)