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Giải bài tập 3.31 trang 71 SGK Toán 9 tập 1 - Cùng khám phá

Cho biểu thức: \(P = \left( {1 + \frac{{x + \sqrt x }}{{\sqrt x + 1}}} \right)\left( {1 - \frac{{x - \sqrt x }}{{\sqrt x - 1}}} \right)\) với \(x \ge 0\) và \(x \ne 1\). a) Rút gọn P. b) Tính giá trị của P khi \(x = 5\).

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Cho biểu thức: \(P = \left( {1 + \frac{{x + \sqrt x }}{{\sqrt x  + 1}}} \right)\left( {1 - \frac{{x - \sqrt x }}{{\sqrt x  - 1}}} \right)\) với \(x \ge 0\) và \(x \ne 1\). a) Rút gọn P. b) Tính giá trị của P khi \(x = 5\).

Phương pháp giải - Xem chi tiết

a) Ta có: \(1 + \frac{{x + \sqrt x }}{{\sqrt x  + 1}} = 1 + \sqrt x ,1 - \frac{{x - \sqrt x }}{{\sqrt x  - 1}} = 1 - \sqrt x \), từ đó rút gọn được P. b) Thay \(x = 5\) vào biểu thức P để tính giá trị biểu thức.

Lời giải chi tiết

a) Với \(x \ge 0\) và \(x \ne 1\) ta có: \(P = \left( {1 + \frac{{x + \sqrt x }}{{\sqrt x  + 1}}} \right)\left( {1 - \frac{{x - \sqrt x }}{{\sqrt x  - 1}}} \right)\)\( = \left( {1 + \frac{{\sqrt x \left( {\sqrt x  + 1} \right)}}{{\sqrt x  + 1}}} \right)\left( {1 - \frac{{\sqrt x \left( {\sqrt x  - 1} \right)}}{{\sqrt x  - 1}}} \right)\)\( = \left( {1 + \sqrt x } \right)\left( {1 - \sqrt x } \right) = 1 - x\) b) Với \(x = 5\) (thỏa mãn điều kiện) thay vào P ta có: \(P = 1 - 5 =  - 4\).

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