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Lý thuyết Định lí Viète và ứng dụng Toán 9 Cánh diều

1. Định lí Viète Nếu \({x_1},{x_2}\) là hai nghiệm của phương trình \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\) thì \({x_1} + {x_2} = - \frac{b}{a}\); \({x_1}{x_2} = \frac{c}{a}.\)

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

Toán - Văn - Anh
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1. Định lí Viète

Nếu \({x_1},{x_2}\) là hai nghiệm của phương trình \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\) thì \({x_1} + {x_2} =  - \frac{b}{a}\); \({x_1}{x_2} = \frac{c}{a}.\)

Ví dụ: Phương꧟ trình \(2{x^2} + 11x + 7 = 0\) có: \💜(\Delta  = {11^2} - 4.2.7 = 65 > 0\) nên phương trình có hai nghiệm phân biệt \({x_1},{x_2}\).

Theo định lí Viète, ta có: \({x_1} + {x_2} =  - \frac{{11}}{2};{x_1}{x_2} = \frac{7}{2}\).

Nhận xét:

Xét phương trình \(a{x^2} + bx + c = 0\) \(\left( {a \ne 0} \right)\): - Nếu \(ac < 0\) thì \(\Delta  = {b^2} - 4ac > 0\), do đó phương trình luôn có hai nghiệm phân biệt. - Nếu \(a + b + c = 0\) thì phương trình có một nghiệm là \({x_1} = 1\), còn nghiệm kia là \({x_2} = \frac{c}{a}\). - Nếu \(a - b + c = 0\) thì phương trình có nghiệm là \({x_1} =  - 1\), còn nghiệm kia là \({x_2} =  - \frac{c}{a}\).

Ví dụ: Phương trình \({x^2} + 3572x - 3573 = 0\) có \(a = 1 > 0,c =  - 3573 &lt; 0\), suy ra a và c 🌄trái dấu. Do đó phương trình có h✤ai nghiệm phân biệt.

Phương trình \({x^2} - 6x + 5 = 0\) có \(a + b + c = 1 + \left( { - 6} \right) + 5 = 0\) nên phương trình có hai nghiệm: \({x_1} = 1,{x_2} = 5\).Phương trình \(5{x^2} + 14x + 9 = 0\) có \(a - b + c = 5 - 14 + 9 = 0\) nên phương trình có hai nghiệm: \({x_1} =  - 1,{x_2} =  - \frac{9}{5}\).

2. Tìm hai số khi biết tổng và tích

Nếu hai số có tổng bằng S và tích bằng P thì hai số đó là hai nghiệm của phương trình bậc hai: \({x^2} - Sx + P = 0\). Điều kiện để có hai số đó là \({S^2} - 4P \ge 0\).

Ví dụ: Hai số có tổng bằng 9, tích bằng 20 là nghiệ𝕴m của phương trình \({x^2} + 9x + 20 = 0\).

Ta có: \(\Delta  = {\left( { - 9} \right)^2} - 4.1.20 = 1,\sqrt \Delta   = 1\).Suy ra phương trình có hai nghiệm \({x_1} = \frac{{9 - 1}}{2} = 4;{x_2} = \frac{{9 + 1}}{2} = 5\).Vậy hai số cần tìm là 4 và 5.

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