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algebra难度:基础algebraquadraticdiscriminant
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判别式pànbiéshì

biệt thức
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更新于 2025-01-24
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Khai niem co ban

Biet thuc (ky hieu Δ\Delta) la mot gia tri quan trong duoc tinh tu cac he so cua phuong trinh bac hai, giup xac dinh tinh chat va so luong nghiem cua phuong trinh.

Dinh nghia

Cho phuong trinh bac hai dang chuan:

ax2+bx+c=0(a0)ax^2 + bx + c = 0 \quad (a \neq 0)

biet thuc duoc dinh nghia la:

Δ=b24ac\Delta = b^2 - 4ac

Bieu thuc nay xuat hien duoi dau can trong cong thuc nghiem:

x=b±Δ2a=b±b24ac2ax = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Ba truong hop

Truong hop 1: Δ>0\Delta > 0 — Hai nghiem thuc phan biet

Phuong trinh co hai nghiem thuc khac nhau:

x1=b+Δ2a,x2=bΔ2ax_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}

Vi du: x25x+6=0x^2 - 5x + 6 = 0

Δ=(5)24(1)(6)=2524=1>0\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0

x1=5+12=3,x2=512=2x_1 = \frac{5 + 1}{2} = 3, \quad x_2 = \frac{5 - 1}{2} = 2

Truong hop 2: Δ=0\Delta = 0 — Nghiem kep (nghiem lap)

Phuong trinh co dung mot nghiem thuc (nghiem kep):

x1=x2=b2ax_1 = x_2 = \frac{-b}{2a}

Vi du: x26x+9=0x^2 - 6x + 9 = 0

Δ=(6)24(1)(9)=3636=0\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0

x=62=3x = \frac{6}{2} = 3

Truong hop 3: Δ<0\Delta < 0 — Khong co nghiem thuc

Phuong trinh khong co nghiem thuc (hai nghiem phuc lien hop).

Vi du: x2+2x+5=0x^2 + 2x + 5 = 0

Δ=224(1)(5)=420=16<0\Delta = 2^2 - 4(1)(5) = 4 - 20 = -16 < 0

Khong co nghiem thuc.

Giai thich bang do thi

Biet thuc xac dinh cach parabol y=ax2+bx+cy = ax^2 + bx + c cat truc x:

Biet thucParabol va truc xSo diem cat
Δ>0\Delta > 0Cat truc x tai hai diem2
Δ=0\Delta = 0Tiep xuc truc x tai mot diem (dinh)1
Δ<0\Delta < 0Khong cat truc x0

Neu a>0a > 0 (parabol mo len tren) va Δ<0\Delta < 0, toan bo parabol nam phia tren truc x, tuc la ax2+bx+c>0ax^2 + bx + c > 0 voi moi xx thuc.

Bai tap thuc hanh CSCA

💡 Luu y: Cac bai tap thuc hanh sau day duoc thiet ke dua tren chuong trinh thi CSCA va dinh dang bai thi tieu chuan cua Trung Quoc de giup hoc sinh lam quen voi cac loai cau hoi va phuong phap giai quyet van de.

Bai 1: Co ban (Do kho ★★☆☆☆)

Xac dinh biet thuc va tinh chat nghiem cho: 2x2+3x2=02x^2 + 3x - 2 = 0

Giai:

Δ=324(2)(2)=9+16=25>0\Delta = 3^2 - 4(2)(-2) = 9 + 16 = 25 > 0

Vi Δ>0\Delta > 0, phuong trinh co hai nghiem thuc phan biet.

x=3±254=3±54x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}

x1=12,x2=2x_1 = \frac{1}{2}, \quad x_2 = -2


Bai 2: Trung cap (Do kho ★★★☆☆)

Voi gia tri nao cua mm thi phuong trinh x2+mx+4=0x^2 + mx + 4 = 0 khong co nghiem thuc?

Giai:

Khong co nghiem thuc co nghia la Δ<0\Delta < 0:

Δ=m24(1)(4)<0\Delta = m^2 - 4(1)(4) < 0 m216<0m^2 - 16 < 0 m2<16m^2 < 16 4<m<4-4 < m < 4

Dap an: 4<m<4-4 < m < 4


Bai 3: Nang cao (Do kho ★★★★☆)

Chung minh rang phuong trinh (k+1)x2+2kx+(k1)=0(k+1)x^2 + 2kx + (k-1) = 0 luon co nghiem thuc voi moi gia tri thuc cua kk (voi k1k \neq -1).

Giai:

Δ=(2k)24(k+1)(k1)\Delta = (2k)^2 - 4(k+1)(k-1) =4k24(k21)= 4k^2 - 4(k^2 - 1) =4k24k2+4= 4k^2 - 4k^2 + 4 =4>0= 4 > 0

Vi Δ=4>0\Delta = 4 > 0 voi moi kk, phuong trinh luon co hai nghiem thuc phan biet. \blacksquare

Loi thuong gap

❌ Loi 1: Sai dau trong b24acb^2 - 4ac

Sai: Δ=b2+4ac\Delta = b^2 + 4ac

Dung: Δ=b24ac\Delta = b^2 - 4ac

❌ Loi 2: Quen dau ngoac khi bb am

Sai: Voi x26x+5=0x^2 - 6x + 5 = 0: Δ=624(1)(5)=3620\Delta = -6^2 - 4(1)(5) = -36 - 20

Dung: Δ=(6)24(1)(5)=3620=16\Delta = (-6)^2 - 4(1)(5) = 36 - 20 = 16

❌ Loi 3: Nham lan giua Δ=0\Delta = 0 va Δ<0\Delta < 0

Sai: Δ=0\Delta = 0 co nghia la khong co nghiem ✗

Dung: Δ=0\Delta = 0 co nghia la nghiem kep; Δ<0\Delta < 0 co nghia la khong co nghiem thuc ✓

Meo hoc tap

  1. Thuoc long cong thuc: Δ=b24ac\Delta = b^2 - 4ac — day la nen tang cua viec phan tich nghiem.
  2. Chu y dau hieu: Dac biet khi bb am, dung quen dau ngoac.
  3. Lien he voi do thi: Luon lien ket biet thuc voi hanh vi giao diem cua parabol voi truc x.
  4. Luyen tap cau hoi tham so: Cau hoi CSCA thuong yeu cau xac dinh pham vi tham so de dat duoc loai nghiem nhat dinh.

💡 Meo thi: Biet thuc la cong cu trung tam trong dai so CSCA. Cac cau hoi voi tham so (kk, mm) va dieu kien nghiem xuat hien rat thuong xuyen — hay luyen tap cac bat phuong trinh lien quan den Δ\Delta!