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Lý thuyết Dấu của tam thức bậc hai - SGK Toán 10 Cánh diều

1. Định lí về dấu của tam thức bậc hai

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1. Định lí về dấu của tam thức bậc hai

Cho tam thức bậc hai \(f(x) = a{x^2} + bx + c\) với \(a \ne 0,\Delta  = {b^2} - 4ac.\)+ \(\Delta  < 0\): f(x) cùng dấu với a, \(\forall x \in \mathbb{R}\)+ \(\Delta  = 0\): f(x) cùng dấu với a, \(\forall x \in \mathbb{R}{\rm{\backslash }}\left\{ {\frac{{ - b}}{{2a}}} \right\}\)+ \(\Delta  > 0\): f(x) có 2 nghiệm \({x_1},{x_2}({x_1} < {x_2})\)

 

2. Ví dụ

Xét dấu của tam thức bậc hai: \(f(x) = 2{x^2} + 3x - 2\)Giải:\(\Delta  = {3^2} - 4.2.( - 2) = 25 > 0\)Tam thức bậc hai \(f(x) = 2{x^2} + 3x - 2\) có hai nghiệm phân biệt \({x_1} =  - 2,{x_2} = \frac{1}{2}\) và hệ số \(a = 2 > 0\)Ta có bảng xét dấu \(f(x)\) như sau:

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