判别式pànbiéshì

Khai niem co ban

Biet thuc (ky hieu $\Delta$) 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:

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

biet thuc duoc dinh nghia la:

$\Delta = b^2 - 4ac$

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

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

Ba truong hop

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

Phuong trinh co hai nghiem thuc khac nhau:

$x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

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

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

$x_1 = \frac{5 + 1}{2} = 3, \quad x_2 = \frac{5 - 1}{2} = 2$

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

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

$x_1 = x_2 = \frac{-b}{2a}$

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

$\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0$

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

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

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

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

$\Delta = 2^2 - 4(1)(5) = 4 - 20 = -16 < 0$

Khong co nghiem thuc.

Giai thich bang do thi

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

Biet thuc	Parabol va truc x	So diem cat
$\Delta > 0$	Cat truc x tai hai diem	2
$\Delta = 0$	Tiep xuc truc x tai mot diem (dinh)	1
$\Delta < 0$	Khong cat truc x	0

Neu $a > 0$ (parabol mo len tren) va $\Delta < 0$, toan bo parabol nam phia tren truc x, tuc la $ax^2 + bx + c > 0$ voi moi $x$ 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: $2x^2 + 3x - 2 = 0$

Giai:

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

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

$x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$

$x_1 = \frac{1}{2}, \quad x_2 = -2$

---

Bai 2: Trung cap (Do kho ★★★☆☆)

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

Giai:

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

$\Delta = m^2 - 4(1)(4) < 0$ $m^2 - 16 < 0$ $m^2 < 16$ $-4 < m < 4$

Dap an: $-4 < m < 4$

---

Bai 3: Nang cao (Do kho ★★★★☆)

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

Giai:

$\Delta = (2k)^2 - 4(k+1)(k-1)$ $= 4k^2 - 4(k^2 - 1)$ $= 4k^2 - 4k^2 + 4$ $= 4 > 0$

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

Loi thuong gap

❌ Loi 1: Sai dau trong $b^2 - 4ac$

Sai: $\Delta = b^2 + 4ac$ ✗

Dung: $\Delta = b^2 - 4ac$ ✓

❌ Loi 2: Quen dau ngoac khi $b$ am

Sai: Voi $x^2 - 6x + 5 = 0$: $\Delta = -6^2 - 4(1)(5) = -36 - 20$ ✗

Dung: $\Delta = (-6)^2 - 4(1)(5) = 36 - 20 = 16$ ✓

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

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

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

Meo hoc tap

1. ✅ Thuoc long cong thuc: $\Delta = b^2 - 4ac$ — day la nen tang cua viec phan tich nghiem.
2. ✅ Chu y dau hieu: Dac biet khi $b$ 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 ($k$, $m$) va dieu kien nghiem xuat hien rat thuong xuyen — hay luyen tap cac bat phuong trinh lien quan den $\Delta$!