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Giải bài tập 6.43 trang 30 SGK Toán 9 tập 2 - Kết nối tri thức

Gọi ({x_1},{x_2}) là hai nghiệm của phương trình ({x^2} - 5x + 6 = 0). Khi đó, giá trị của biểu thức (A = x_1^2 + x_2^2) là A. 13. B. 19. C. 25. D. 5.

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

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Đề bài

Gọi \({x_1},{x_2}\) là hai nghiệm của phương trình \({x^2} - 5x + 6 = 0\). Khi đó, giá trị của biểu thức \(A = x_1^2 + x_2^2\) là A. 13. B. 19. C. 25. D. 5.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

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 \(\Delta  > 0\) thì áp dụng định lí Viète để tính tổng và tích các nghiệm \({x_1} + {x_2} = \frac{{ - b}}{a};{x_1}.{x_2} = \frac{c}{a}\). Biến đổi \(x_1^2 + x_2^2 = {\left( {{x_1} + {x_2}} \right)^2} - 2{x_1}{x_2}\), từ đó thay \({x_1} + {x_2} = \frac{{ - b}}{a};{x_1}.{x_2} = \frac{c}{a}\) để tính giá trị biểu thức.

Lời giải chi tiết

Vì \(\Delta  = {\left( { - 5} \right)^2} - 4.6 = 1 > 0\) nên phương trình có hai nghiệm phân biệt. Theo định lí Viète ta có: \({x_1} + {x_2} = 5;{x_1}.{x_2} = 6\) Ta có: \(x_1^2 + x_2^2 = {\left( {{x_1} + {x_2}} \right)^2} - 2{x_1}{x_2} = {5^2} - 2.6 = 13\) Chọn A

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