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Giải bài 2 trang 22 vở thực hành Toán 9 tập 2

Cho phương trình ({x^2} - x - 1 = 0). Không giải phương trình, hãy tính: a) Tổng và tích các nghiệm. b) Tổng các nghịch đảo của các nghiệm.

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

Toán - Văn - Anh
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Đề bài

Cho phương trình \({x^2} - x - 1 = 0\). Không giải phương trình, hãy tính: a) Tổng và tích các nghiệm. b) Tổng các nghịch đảo của các nghiệm.

Phương pháp giải - Xem chi tiết

a) Xét phương trình bậc hai một ẩn \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\). + Tính biệt thức \(\Delta  = {b^2} - 4ac\). + Nếu \(\Delta  \ge 0\) thì áp dụng định lí Viète để tính tổng và tích các nghiệm: \({x_1} + {x_2} = \frac{{ - b}}{a};{x_1}.{x_2} = \frac{c}{a}\) b) Biến đổi \(\frac{1}{{{x_1}}} + \frac{1}{{{x_2}}} = \frac{{{x_1} + {x_2}}}{{{x_1}{x_2}}}\) (*), thay \({x_1} + {x_2} = \frac{{ - b}}{a};{x_1}.{x_2} = \frac{c}{a}\) vào biểu thức (*) để tính.

Lời giải chi tiết

Ta có: \(\Delta  = {\left( { - 1} \right)^2} - 4.1.\left( { - 1} \right) = 5 > 0\). Do đó, phương trình có hai nghiệm \({x_1},{x_2}\). a) Áp dụng định lí Viète ta có: \({x_1} + {x_2} =  - \frac{b}{a} =  - \frac{{ - 1}}{1} = 1;\\{x_1}.{x_2} = \frac{c}{a} = \frac{{ - 1}}{1} =  - 1.\) b) Ta có: \(\frac{1}{{{x_1}}} + \frac{1}{{{x_2}}} = \frac{{{x_1} + {x_2}}}{{{x_1}{x_2}}} = \frac{1}{{ - 1}} =  - 1\).

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