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7. При каких значениях переменной имеет смысл выражениє \( \sqrt{13 x-6 x^{2}-5} \) ?

Ask by Young Luna. in Belarus
Mar 15,2025

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Выражение \(\sqrt{13x-6x^2-5}\) имеет смысл при \( \frac{1}{2} \leq x \leq \frac{5}{3} \).

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Чтобы выражение \( \sqrt{13 x-6 x^{2}-5} \) имело смысл, подкоренное выражение должно быть неотрицательным. Это означает, что \( 13x - 6x^2 - 5 \geq 0 \). Для нахождения допустимых значений переменной можно решить неравенство. Сначала мы найдем корни уравнения \( 6x^2 - 13x + 5 = 0 \) с помощью дискриминанта. Итак, находим \( D = B^2 - 4AC = (-13)^2 - 4 \cdot 6 \cdot 5 = 169 - 120 = 49 \). Корни будут \( x_1 = \frac{13 + 7}{12} = \frac{20}{12} = \frac{5}{3} \) и \( x_2 = \frac{13 - 7}{12} = \frac{6}{12} = \frac{1}{2} \). Таким образом, значения переменной \( x \) лежат в интервале \( \left[\frac{1}{2}, \frac{5}{3}\right] \).

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