2. 判别式 𝒑𝒈 𝟓𝟐, 𝟓𝟑
𝟔. (𝒂) 𝟑𝒙𝟐
+ 𝟓𝒙 − 𝟏 = 𝟎
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
𝒃 𝟓𝒙𝟐 + 𝟕 = 𝟎
𝐚 = 𝟑
𝒃 = 𝟓
𝐜 = −𝟏
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= 𝟓𝟐
− 𝟒 𝟑 −𝟏
= 𝟐𝟓 + 𝟏𝟐
= 𝟑𝟕 > 𝟎
∴ 𝟑𝒙𝟐
+ 𝟓𝒙 − 𝟏 = 𝟎 𝒉𝒂𝒗𝒆 𝟐 𝒅𝒊𝒇𝒇. 𝒓𝒐𝒐𝒕𝒔
𝐚 = 𝟓
𝒃 = 𝟎
𝐜 = 𝟕
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= 𝟎 − 𝟒 𝟓 𝟕
= −𝟏𝟒𝟎 < 𝟎
∴ 𝟓𝒙𝟐 + 𝟕 = 𝟎 𝒉𝒂𝒗𝒆 𝒏𝒐 𝒓𝒆𝒂𝒍 𝒓𝒐𝒐𝒕𝒔
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
3. 判别式 𝒑𝒈 𝟓𝟑
𝟔. 𝒄 𝟒𝒚𝟐
− 𝟏𝟐𝒚 + 𝟗 = 𝟎
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
𝒅 𝟑𝒛𝟐
− 𝟕𝒛 + 𝟑 = 𝟎
𝐚 = 𝟒
𝒃 = −𝟏𝟐
𝐜 = 𝟗
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= −𝟏𝟐 𝟐
− 𝟒 𝟒 𝟗
= 𝟏𝟒𝟒 − 𝟏𝟒𝟒
= 𝟎
∴ 𝟒𝒚𝟐
− 𝟏𝟐𝒚 + 𝟗 = 𝟎 𝒉𝒂𝒗𝒆 𝟐 𝒆𝒒𝒖𝒂𝒍 𝒓𝒐𝒐𝒕𝒔
𝐚 = 𝟑
𝒃 = −𝟕
𝐜 = 𝟑
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= −𝟕 𝟐 − 𝟒 𝟑 𝟑
= 𝟏𝟑 > 𝟎
2 equal roots
no real roots
2 different roots
= 𝟒𝟗 − 𝟑𝟔
∴ 𝟑𝒛𝟐 − 𝟕𝒛 + 𝟑 = 𝟎 𝒉𝒂𝒗𝒆 𝟐 𝒅𝒊𝒇𝒇. 𝒓𝒐𝒐𝒕𝒔
无实根
有两相等实根
有两不等实根
4. 判别式 𝒑𝒈 𝟓𝟒
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
有两不等实根
有两不等实根
有两不等实根
有两不等实根
有两不等实根
有两不等实根
无实根
无实根
无实根
有两相等实根
无实根
有两相等实根
有两不等实根
5. 𝒂 = 𝟏
𝒃 = −(𝟐𝒌 + 𝟏)
𝐜 = (𝒌𝟐 − 𝟏)
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= − 𝟐𝒌 − 𝟏 𝟐
−𝟒 𝟏 𝒌𝟐
− 𝟏
= 𝟒𝒌𝟐 + 𝟒𝒌 + 𝟏 − 𝟒𝒌𝟐 + 𝟒
= 𝟒𝒌 + 𝟓
𝒂 ∆ = 𝟎
𝟒𝒌 + 𝟓 = 𝟎
𝒌 = −
𝟓
𝟒
𝒃 ∆ > 𝟎
𝟒𝒌 + 𝟓 > 𝟎
𝒌 > −
𝟓
𝟒
𝒄 ∆ < 𝟎
𝟒𝒌 + 𝟓 < 𝟎
𝒌 < −
𝟓
𝟒
判别式 𝒑𝒈 𝟓𝟑
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
7
无实根
有两相等实根
有两不等实根
6. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒎 = −
𝟏
𝟒
−
𝟏
𝟒
< 𝒎 < 𝟎 𝒎 > 𝟎
𝒎 < −
𝟏
𝟒
或
𝒑 = 𝟓 𝒙𝟏 = 𝒙𝟐 = −𝟏
7. 𝒂 = 𝟒
𝒃 = −𝟐𝟖
𝐜 = 𝒎
∆ = 𝒃𝟐 − 𝟒𝒂𝒄 = 𝟎
−𝟐𝟖 𝟐 − 𝟒 𝟒 𝒎 = 𝟎
7𝟖𝟒 = 𝟏𝟔𝒎
𝒎 =
𝟕𝟖𝟒
𝟏𝟔
𝒎 = 𝟒𝟗
判别式 𝒑𝒈 𝟓𝟒
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
8
无实根
有两相等实根
有两不等实根
8. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒎 = 𝟎 , 𝟓
𝒎 = 𝟎 , 𝟑
9. 判别式 𝒑𝒈 𝟓𝟒
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
𝒂 = 𝒎 + 𝟑
𝒃 = − 𝒎𝟐 − 𝟖
𝐜 = 𝒎 − 𝟑
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= − 𝒎𝟐 − 𝟖
𝟐
− 𝟒 𝒎 + 𝟑 𝒎 − 𝟑
= −𝒎𝟐
+ 𝟖
𝟐
− 𝟒 𝒎𝟐
− 𝟗
= 𝒎𝟒 −𝟏𝟔𝒎𝟐 + 𝟔𝟒 − 𝟒𝒎𝟐 + 𝟑𝟔
= 𝒎𝟒 −𝟐𝟎𝒎𝟐 + 𝟏𝟎𝟎
= 𝒎𝟐 − 𝟏𝟎
𝟐
无实根
有两相等实根
有两不等实根
10. 𝒂 = 𝒌 + 𝟒
𝒃 = −(𝒌 + 𝟑)
𝐜 = −𝟏
∆ = 𝒃𝟐 − 𝟒𝒂𝒄 ≥ 𝟎
− 𝒌 + 𝟑 𝟐 − 𝟒 𝒌 + 𝟒 −𝟏 ≥ 𝟎
𝒌𝟐 + 𝟗 + 𝟔𝒌 + 𝟒𝒌 + 𝟏𝟔 ≥ 𝟎
𝒌𝟐 + 𝟐𝟓 + 𝟏𝟎𝒌 ≥ 𝟎
𝒌 + 𝟓 𝟐 ≥ 𝟎
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
11. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒎 = 𝟔
12. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
13. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒂 = 𝒑
𝒃 = 𝒑 + 𝒒
𝐜 = 𝒒
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
= 𝒑 + 𝒒 𝟐
− 𝟒 𝒑 𝒒
= 𝒑𝟐
+ 𝟐𝒑𝒒 + 𝒒𝟐
− 𝟒𝒑𝒒
= 𝒑𝟐 − 𝟐𝒑𝒒 + 𝒒𝟐
= 𝒑 − 𝒒 𝟐
𝒑 − 𝒒 𝟐 ≥ 𝟎
∴ ∆ ≥ 𝟎
14. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒂 = 𝟏 − 𝒒 +
𝒑𝟐
𝟐
𝒃 = 𝒑(𝟏 + 𝒒)
𝐜 = 𝒒 𝒒 − 𝟏 +
𝒑𝟐
𝟐
∆ = 𝒃𝟐
− 𝟒𝒂𝒄 ≥ 𝟎
∆ = 𝒑 𝟏 + 𝒒 𝟐 − 𝟒 𝟏 − 𝒒 +
𝒑𝟐
𝟐
𝒒 𝒒 − 𝟏 +
𝒑𝟐
𝟐
∆ 𝒘𝒊𝒍𝒍 𝒐𝒏𝒍𝒚 𝒃𝒆 𝒕𝒓𝒖𝒆 𝒊𝒇 𝒃𝟐 = 𝟒𝒂𝒄
𝑳𝒆𝒕 𝒑𝟐
= 𝟒𝒒
𝟒𝒒 𝟏 + 𝟐𝒒 + 𝒒𝟐
= 𝟒 𝟏 − 𝒒 +
𝟒𝒒
𝟐
𝒒 𝒒 − 𝟏 +
𝟒𝒒
𝟐
= 𝟒 𝟏 − 𝒒 + 𝟐𝒒 (𝒒𝟐 − 𝒒 + 𝟐𝒒)
= 𝟒 𝟏 + 𝒒 (𝒒𝟐
+ 𝒒)
= 𝟒 𝒒𝟐 + 𝒒 + 𝒒𝟑 + 𝒒𝟐
= 𝟒𝒒 𝟏 + 𝟐𝒒 + 𝒒𝟐
15. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒂𝟐 + 𝒃𝟐 − 𝒄𝟐 = 𝟎
𝒄 − 𝒂 𝟐 = 𝟒(𝒃 − 𝒄)(𝒂 − 𝒃)
16. 判别式 𝒑𝒈 𝟓𝟓
∆ = 𝒃𝟐 − 𝟒𝒂𝒄
2 equal roots
no real roots
2 different roots
无实根
有两相等实根
有两不等实根
𝒂𝟐 + 𝒃𝟐 − 𝒄𝟐 = 𝟎
𝒂(𝟏 − 𝒙𝟐) + 𝟐𝒃𝒙 + 𝒄 𝟏 + 𝒙𝟐 = 𝟎
𝒂 − 𝒂𝒙𝟐 + 𝟐𝒃𝒙 + 𝒄 + 𝒄𝒙𝟐 = 𝟎
(−𝒂 + 𝒄)𝒙𝟐
+ 𝟐𝒃𝒙 + 𝒂 + 𝒄 = 𝟎
𝐚 = −𝒂 + 𝒄
𝒃 = 𝟐𝒃
𝐜 = 𝒂 + 𝒄
∆ = 𝒃𝟐
− 𝟒𝒂𝒄 = 𝟎
𝟐𝒃 𝟐 − 𝟒 −𝒂 + 𝒄 𝒂 + 𝒄 = 𝟎
𝟒𝒃𝟐 − 𝟒 𝒄𝟐 − 𝒂𝟐 = 𝟎
