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Giải bài 3 trang 48 SGK Toán 10 tập 1 – Cánh diều

Xét dấu của mỗi tam thức bậc hai sau:

🔯Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Xét dấu của mỗi tam thức bậc hai sau: a) \(f\left( x \right) = 3{x^2} - 4x + 1\) b) \(f\left( x \right) = 9{x^2} + 6x + 1\) c) \(f\left( x \right) = 2{x^2} - 3x + 10\) d) \(f\left( x \right) =  - 5{x^2} + 2x + 3\) e) \(f\left( x \right) =  - 4{x^2} + 8x - 4\) g) \(f\left( x \right) =  - 3{x^2} + 3x - 1\)

Phương pháp giải - Xem chi tiết

Sử dụng biệt thức thu gọn \(\Delta ' = {\left( {b'} \right)^2} - ac\) với \(b = 2b'\). + Nếu \(\Delta ' < 0\) thì \(f\left( x \right)\) cùng dấu với hệ số a vời mọi \(x \in \mathbb{R}\). + Nếu \(\Delta ' = 0\) thì \(f\left( x \right)\) cùng dấu với hệ số a vời mọi \(x \in \mathbb{R}\backslash \left\{ { - \frac{{b'}}{a}} \right\}\). + Nếu \(\Delta ' > 0\) thì \(f\left( x \right)\) có 2 nghiệm \({x_1},{x_2}\left( {{x_1} < {x_2}} \right)\). Khi đó: \(f\left( x \right)\) cùng dấu với hệ số a với mọi x thuộc các khoảng \(\left( { - \infty ;{x_1}} \right)\) và \(\left( {{x_2}; + \infty } \right)\); \(f\left( x \right)\) trái dấu với hệ số a với mọi x thuộc các khoảng \(\left( {{x_1};{x_2}} \right)\)

Lời giải chi tiết

a) Ta có \(a = 3 > 0,b =  - 4,c = 1\)\(\Delta ' = {\left( { - 2} \right)^2} - 3.1 = 1 > 0\)\( \Rightarrow \)\(f\left( x \right)\) có 2 nghiệm \(x = \frac{1}{3},x = 1\). Khi đó:\(f\left( x \right) > 0\) với mọi x thuộc các khoảng \(\left( { - \infty ;\frac{1}{3}} \right)\) và \(\left( {1; + \infty } \right)\);\(f\left( x \right) < 0\) với mọi x thuộc các khoảng \(\left( {\frac{1}{3};1} \right)\)b) Ta có \(a = 9 > 0,b = 6,c = 1\)\(\Delta ' = 0\)\( \Rightarrow \)\(f\left( x \right)\) có 1 nghiệm \(x =  - \frac{1}{3}\). Khi đó:\(f\left( x \right) > 0\) với mọi \(x \in \mathbb{R}\backslash \left\{ { - \frac{1}{3}} \right\}\)c) Ta có \(a = 2 > 0,b =  - 3,c = 10\)\(\Delta  = {\left( { - 3} \right)^2} - 4.2.10 =  - 71 < 0\)\( \Rightarrow \)\(f\left( x \right) > 0\forall x \in \mathbb{R}\)d) Ta có \(a =  - 5 < 0,b = 2,c = 3\)\(\Delta ' = {1^2} - \left( { - 5} \right).3 = 16 > 0\)\( \Rightarrow \)\(f\left( x \right)\) có 2 nghiệm \(x = \frac{{ - 3}}{5},x = 1\). Khi đó:\(f\left( x \right) < 0\) với mọi x thuộc các khoảng \(\left( { - \infty ; - \frac{3}{5}} \right)\) và \(\left( {1; + \infty } \right)\);
\(f\left( x \right) > 0\) với mọi x thuộc các khoảng \(\left( { - \frac{3}{5};1} \right)\)e) Ta có \(a =  - 4 < 0,b = 8c =  - 4\)\(\Delta ' = 0\)\( \Rightarrow \)\(f\left( x \right)\) có 1 nghiệm \(x = 1\). Khi đó:\(f\left( x \right) < 0\) với mọi \(x \in \mathbb{R}\backslash \left\{ 1 \right\}\)g) Ta có \(a =  - 3 < 0,b = 3,c =  - 1\)\(\Delta  = {3^2} - 4.\left( { - 3} \right).\left( { - 1} \right) =  - 3 < 0\)\( \Rightarrow \)\(f\left( x \right) < 0\forall x \in \mathbb{R}\)

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Ph/hs Tham Gia Nhóm Để Cập Nhật Điểm Thi, Điểm Chuẩn Miễn Phí

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