TRƯỜNG THCS PHÚ THỌ QUẬN 11 GV: CAO MINH TÀI TRANG 1 CÁC BÀI TẬP MẪU 1/ x(x–1)(2–3x) = 0 ⇔ x = 0 hay x– 1 = 0 hay 2–3x = 0 ⇔ x= 0 hay x = 1 hay –3x = –2 ⇔ x = 0 hay x = 1 hay x = 2 3 2/ ( 1 2 x +1)(9 x 2 –25 ) = 0 ⇔ ( 1 2 x +1)(3x–5)(3x+5) = 0 ⇔ 1 2 x = –1 hay 3x = 5 hay 3x = –5 ⇔ x = –2 hay x = 5 3 hay x = −5 3 3/ x( x 3 –2)( x 2 +3) = 0 ⇔ x = 0 hay x 3 –2 = 0 hay x 2 +3 = 0 ⇔ x= 0 hay x 3 = 2 hay x 2 =–3 (vô lí) ⇔ x = 0 hay x = 6 4/ ( x 2 +2)( x 2 –2x+1) = 0 ⇔ ( x 2 +2)(x–1) 2 = 0 ⇔ x 2 +2 = 0 hay (x–1) 2 = 0 ⇔ x 2 = –2 (vô lí) hay x– 1 = 0 ⇔ x = 1 5/ 2x(x +3) – 4(x +3) = 0 ⇔ (x+3)(2x–4) = 0 ⇔ x+3=0 hay 2x–4=0 ⇔ x = –3 hay x = 2 6/ x 2 (x+1) – 4(x+1)=0 ⇔ (x+1)( x 2 –4) = 0 ⇔ (x+1)(x–2)(x+2) = 0 ⇔ x= –1 hay x = 2 hay x = –2 7/ (2x–5)(x+4) –(x+4)(x–3) = 0 ⇔ (x+4).[(2x–5)–(x–3)]=0 ⇔ (x+4)(2x–5–x+3) =0 ⇔ (x+4)(x–2) =0 ⇔ x = –4 hay x =2 8/ (x–5) 2 –2(x–5) =0 ⇔ (x–5)(x–5)–2(x–5)=0 ⇔ (x–5)[(x–5)–2] =0 ⇔ (x–5)(x–7)=0 ⇔ x=5 hay x = 7 9/ 5x(x+1)–7(1+x) = 0 ⇔ 5x(x+1)–7(x+1)= 0 ⇔ (x+1)(5x–7) = 0 ⇔ x = –1 hay x = 7 5 10/ 2x(5x–2)–3(2–5x) = 0 ⇔ 2x(5x–2)+3(5x–2)= 0 ⇔ (5x–2)(2x+3)=0 ⇔ x = 2 5 hay x = −3 2 25/ 4 x 2 –4x+1 = (x–3) 2 11/ x(x–2)–3x+6 = 0 ⇔ x(x–2)–(3x–6) = 0 ⇔ x(x–2)–3(x–2) = 0 ⇔ (x–2)(x–3) = 0 ⇔ x = 2 hay x = 3 12/ 6(x+2) – x 2 +4 = 0 ⇔ 6(x+2) –( x 2 –4) = 0 ⇔ 6(x+2) –(x+2)(x–2) = 0 ⇔ (x+2)[6–(x–2)] = 0 ⇔ (x+2)(6–x+2) = 0 ⇔ (x+2)(8–x)=0 ⇔ x = –2 hay x = 8 13/ x 3 +2 x 2 –x–2 = 0 ⇔ ( x 3 +2 x 2 )–(x+2) = 0 ⇔ x 2 (x+2)–(x+2) = 0 ⇔ (x+2)( x 2 –1)= 0 ⇔ (x+2)(x–1)(x+1)=0 ⇔ x =–2 hay x = 1 hay x = –1 14/ x 2 =2x ⇔ x 2 –2x = 0 ⇔ x(x–2) = 0 ⇔ x = 0 hay x–2 = 0 ⇔ x = 0 hay x = 2 15/ (3x+2)(x–4)=2x(x–4) ⇔ (3x+2)(x–4)–2x(x–4) = 0 ⇔ (x–4)[(3x+2)–2x] = 0 ⇔ (x–4)(3x+2–2x) =0 ⇔ (x–4)(x+2)=0 ⇔ x= 4 hay x = –2 16/ 5(x+3)(x–2)= 3(x+5)(x–2) ⇔ 5(x+3)(x–2)– 3(x+5)(x–2) = 0 ⇔ (x–2)[5(x+3)–3(x+5)] =0 ⇔ (x–2)(5x+15–3x–15) = 0 ⇔ (x–2)(2x) = 0 ⇔ x–2 = 0 hay 2x = 0 ⇔ x = 2 hay x = 0 17/ (x–1)(2x–1)=x(1–x) ⇔ (x–1)(2x–1)–x(1–x) = 0 ⇔ (x–1)(2x–1) +x(x–1) = 0 ⇔ (x–1)(2x–1+x) = 0 ⇔ (x–1)(x–1)=0 ⇔ x–1 = 0 ⇔ x = 1 18/ (2x+2)(x–3)= 3(x+1) ⇔ 2(x+1)(x–3)= 3(x+1) ⇔ 2(x+1)(x–3)– 3(x+1) = 0 ⇔ (x+1)[2(x–3)–3] = 0 ⇔ (x+1)(2x–6–3) = 0 ⇔ (x+1)(2x–9) = 0 ⇔ x = –1 hay x = 4,5 34/ 2x – 5 < 5x –3 19/ 3x–12 = 5x(x–4) ⇔ 3(x–4) = 5x(x–4) ⇔ 3(x–4)– 5x(x–4) = 0 ⇔ (x–4)(3–5x)=0 ⇔ x = 4 hay x = 3 5 20/ x 2 –4 = 0 ⇔ (x–2)(x+2) = 0 ⇔ x–2 = 0 hay x+2 = 0 ⇔ x = 2 hay x = –2 21/ *Cách 1: 16 x 2 = 1 ⇔ x 2 = 1 16 ⇔ x = ± 1 4 *Cách 2: 16 x 2 = 1 ⇔ 16 x 2 –1 = 0 ⇔ (4x) 2 –1 2 = 0 ⇔ (4x–1)(4x+1) = 0 ⇔ 4x– 1 = 0 hay 4x+1 = 0 ⇔ x = 1 4 hay x = −1 4 22/ *Cách 1: 8 x 2 = 2 ⇔ x 2 = 2 8 ⇔ x 2 = 1 4 ⇔ x = ± 1 2 *Cách 2: 8 x 2 = 2 ⇔ 8 x 2 –2 = 0 ⇔ 2(4 x 2 –1) = 0 ⇔ 2(2x–1)(2x+1) = 0 ⇔ 2x–1 = 0 hay 2x+1 = 0 ⇔ x = 1 2 hay x = −1 2 23/ (2x–1) 2 = 9x 2 ⇔ (2x–1) 2 –9x 2 = 0 ⇔ (2x–1) 2 –(3x) 2 = 0 ⇔ [(2x–1)–3x][(2x–1)+3x] = 0 ⇔ (–x–1)(5x–1) = 0 ⇔ –x–1 = 0 hay 5x–1 = 0 ⇔ x = –1 hay x = 1 5 24/ (5x–3) 2 –(4x–7) 2 = 0 ⇔[(5x–3)–(4x–7)][(5x–3)+(4x–7)]=0 ⇔ (5x–3–4x+7)(5x–3+4x–7) = 0 ⇔ (x+4)(9x–10) = 0 ⇔ x+4 = 0 hay 9x– 10 = 0 ⇔ x = –4 hay x = 10 9 41/ x x+ − < 2 3 2 3 2 TRƯỜNG THCS PHÚ THỌ QUẬN 11 GV: CAO MINH TÀI TRANG 2 ⇔ (2x–1) 2 –(x–3) 2 = 0 ⇔[(2x–1)–(x–3)][(2x–1)+(x–3)]= 0 ⇔ (2x–1–x+3)(2x–1+x–3) = 0 ⇔ (x+2)(3x–4) = 0 ⇔ x+2 = 0 hay 3x–4 = 0 ⇔ x = –2 hay x = 4 3 26/ (2x+7) 2 –9(x+2) 2 = 0 ⇔ (2x+7) 2 –3 2 .(x+2) 2 = 0 ⇔ (2x+7) 2 – [ 3(x+2) ] 2 = 0 ⇔ (2x+7) 2 –(3x+6) 2 = 0 ⇔(–x+1)(5x+13) =0 ⇔ x = 1 hay x = −13 5 27/ 4 x 2 +4x+1 = 0 ⇔ (2x+1) 2 = 0 ⇔ 2x+1 = 0 ⇔ x = −1 2 28/ 16 x 2 +8x = –1 ⇔ 16 x 2 +8x +1 = 0 ⇔ (4x+1) 2 = 0 ⇔ 4x+1 = 0 ⇔ x = −1 4 29/ x 2 +3x–4 = 0 ⇔ x 2 –x+4x – 4 = 0 ⇔ ( x 2 –x)+(4x – 4) = 0 ⇔ x(x – 1)+4(x–1) = 0 ⇔ (x–1)(x+4) = 0 ⇔ x = 1 hay x = –4 30/ 2x–7 ≤ 0 ⇔ 2x ≤ 7 ⇔ x ≤ 7 2 ⇔ x ≤ 3,5 31/ –3x + 9 ≥ 0 ⇔ –3x ≥ –9 ⇔ x ≤ − − 9 3 ⇔ x ≤ 3 32/ 15–3x > 9 ⇔ –3x > 9–15 ⇔ –3x > –6 ⇔ x < − − 6 3 ⇔ x < 2 33/ –2 > 1– 3x ⇔ 3x > 1+2 ⇔ 3x > 3 ⇔ x > 1 ⇔ 2x – 5x < –3 +5 ⇔ –3x < 2 ⇔ x > − 2 3 ⇔ x > −2 3 35/ 1 + 2(x–1) > 3– 2x ⇔ 1+2x – 2 > 3– 2x ⇔ 2x + 2x > 3–1 +2 ⇔ 4x > 4 ⇔ x > 1 36/ x– 8 ≥ 2(x + 1 2 ) + 7 ⇔ x– 8 ≥ 2x + 1 + 7 ⇔ x– 2x ≥ 1 +7 + 8 ⇔ –x ≥ 16 ⇔ x ≤ –16 37/ (x–3) 2 < x 2 –3 ⇔ x 2 – 6x+ 9 < x 2 –3 ⇔ x 2 – x 2 –6x < –3 – 9 ⇔ –6x < –12 ⇔ x > − − 12 6 ⇔ x > 2 *Cách 2 : (x–3) 2 < x 2 –3 ⇔ (x–3)(x–3) < x 2 –3 ⇔ x 2 –3x–3x + 9 < x 2 –3 ⇔ x 2 – x 2 –3x–3x < –3 –9 ⇔ –6x < –12 ⇔ x > 2 38/ x(x–3) – (x–2) 2 < 0 ⇔ x 2 –3x – [ x 2 –4x +4 ] < 0 ⇔ x 2 –3x – x 2 + 4x – 4 < 0 ⇔ x < 4 39/ (x–2) 2 ≥ 3– (4–x)(x+5) ⇔ x 2 –4x+4 ≥ 3– [4x+20– x 2 –5x] ⇔ x 2 –4x+4 ≥ 3– 4x–20 + x 2 +5x ⇔ x 2 – x 2 –4x +4x –5x ≥ 3–20–4 ⇔ –5x ≥ –21 ⇔ x ≤ − − 21 5 ⇔ x ≤ 21 5 40/ 2(2x–1) 2 + 6 ≥ 8(x+3)(x–3) ⇔ 2(4 x 2 –4x+1) + 6 ≥ 8( x 2 –9) ⇔ 8 x 2 –8x + 2 + 6 ≥ 8 x 2 – 72 ⇔ 8 x 2 – 8 x 2 –8x ≥ –72 –2 – 6 ⇔ –8x ≥ –80 ⇔ x ≤ − − 80 8 ⇔ x ≤ 10 ⇔ 2(2+3x) < 3(x–2) ⇔ 4 + 6x < 3x– 6 ⇔ 6x – 3x < –6 –4 ⇔ 3x < –10 ⇔ x < −10 3 42/ x − > 3 1 2 4 ⇔ 3x– 1 > 4 . 2 ⇔ 3x > 8 + 1 ⇔ 3x > 9 ⇔ x > 3 43/ x− < 1 5 3 ⇔ –x < 3 . 5 ⇔ –x < 15 ⇔ x > –15 44/ x− < 2 2 0 3 ⇔ 2–2x < 0 . 3 ⇔ 2–2x < 0 ⇔ –2x < –2 ⇔ x > − − 2 2 ⇔ x > 1 45/ x < − 2 0 3 ⇔ 3–x < 0 ⇔ –x < –3 ⇔ x > 3 46/ x − < − 2 0 3 1 ⇔ 3x– 1 > 0 ⇔ 3x > 1 ⇔ x > 1 3 47/ x x x− − − − − ≥ 1 2 13 5 3 3 12 4 (MTC:12) ⇔ 4(1–2x)–1(13–5x) ≥ 3(–x–3) ⇔ 4–8x –13 +5x ≥ –3x –9 ⇔ –8x + 5x +3x ≥ –9 –4 +13 ⇔ 0x ≥ 0 ⇔ 0 ≥ 0 (đúng) Vậy bất phương trình trên vô số nghiệm 48/ x x − − > − 2 2 1 2 3 ⇔ x x− − − > 2 2 1 2 3 1 (MTC: 6) ⇔ 3(x–2) –4 > 6(x–1) ⇔ 3x – 6 –4 > 6x – 6 ⇔ 3x – 6x > –6 + 6 +4 ⇔ –3x > 4 ⇔ x < − 4 3 ⇔ x < −4 3 Giải các phương trình sau: TRƯỜNG THCS PHÚ THỌ QUẬN 11 GV: CAO MINH TÀI TRANG 3 49/ x− =1 2 5 x x (dung) hay ≥  ⇔  − = − = −  5 0 1 2 5 1 2 5 x x⇔ − = − − = − −2 5 1 2 5 1 hay x x⇔ − = − = −2 4 2 6 hay x x⇔ = − =2 3 hay Vậy S= { } ;−2 3 50/ x − = −3 5 4 Vì –4 < 0 nên phương trình trên vô nghiệm 51/ x − − = −1 2 9 4 x ⇔ − = − +1 2 4 9 x ⇔ − =1 2 5 (câu 49) 52/ x x− =2 5 3 x x x x x hay ≥  ⇔  − = − = −  3 0 2 5 3 2 5 3 x x x x x ≥  ⇔  − = + =  0 2 3 5 2 3 5 hay x x x ≥  ⇔  − = =  0 5 5 5 hay x x (loai) x (nhân) ≥  ⇔  = − =  0 5 1 hay Vậy S= { } 1 53/ x x− + =4 3 2 x x⇔ − = −4 3 2 x x x x x − ≥  ⇔  − = − − = − +  2 0 4 3 2 4 3 2 hay x x x x x hay ≥  ⇔  − − = − − − + = −  2 3 2 4 3 2 4 x x x ≥  ⇔  − = − − = −  2 4 6 2 2 hay x x x (loai) hay 1 (loai) ≥   ⇔  = =   2 3 2 Vậy S= ∅ 54/ x + + =2 1 1 9 x⇔ + = −2 1 9 1 x⇔ + =2 1 8 x⇔ + = 8 1 2 x⇔ + =1 4 x x (dung) hay ≥  ⇔  + = + = −  4 0 1 4 1 4 x x hay ⇔ = = −3 5 Vậy S= { } ;−3 5 55/ x x + = − 2 1 1 2 (1) * MTC: (x–2) *ĐKXĐ: x ≠ 2 * QĐ: (1) ⇔ 2x+1 = x–2 ⇔ 2x– x = –2 –1 ⇔ x = –3 (nhận) Vậy S= { } −3 56/ x x x x − + = + − − 2 1 1 3 12 2 2 4 x x x (x )(x ) (1) − ⇔ − = + − − + 1 1 3 12 2 2 2 2 * MTC: (x+2)(x–2) *ĐKXĐ: x ≠ –2; x ≠ 2 * QĐ: (1) ⇔ 1(x–2) – 1(x+2) = 3x–12 ⇔ x– 2 –x –2 = 3x– 12 ⇔ –3x = –8 ⇔ x = 8 3 (nhận) Vậy S=       8 3 57/ x x x (x ) x (x )(x ) + = − + + − 2 2 3 2 2 1 3 x x x (x ) (x ) (x )(x ) ⇔ + = − + + − 2 2 3 2 1 1 3 (1) * MTC: 2(x–3)(x+1) *ĐKXĐ: x ≠ 3; x ≠ –1 * QĐ: (1)⇔ x(x+1) + x(x–3) = 4x ⇔ x 2 + x + x 2 –3x –4x = 0 ⇔ 2 x 2 –6x = 0 ⇔ 2x(x – 3) = 0 ⇔ 2x = 0 hay x– 3 = 0 ⇔ x = 0 (nhận) hay x = 3 (loại) Vậy S= { } 0 58/ x (x ) (x ) (x )(x ) − − = + − − + 3 1 2 2 1 6 1 1 1 * MTC: 6(x–1)(x+1) *ĐKXĐ: x ≠ 1; x ≠ –1 * QĐ: (1) ⇔ 9(x–1) – 1(x+1) = –12x ⇔ 9x – 9 –x –1 = –12x ⇔ 20x = 10 ⇔ x = 1 2 (nhận) Vậy S=       1 2 59/ x x x x x x x x x − + + + = + − − 2 2 3 2 3 1 6 2 3 2 3 4 9 x x x x( x ) x( x ) x( x ) − + + ⇔ + = + − − 2 2 3 1 6 2 3 2 3 4 9 x x x x( x ) x( x ) x( x )( x ) − + + ⇔ + = + − − + 2 3 1 6 2 3 2 3 2 3 2 3 * MTC: x(2x+3)(2x–3) *ĐKXĐ: x ≠ 0; x ≠ −3 2 ; x ≠ 3 2 * QĐ: (1) ⇔ (2x–3) 2 +(x+1)(2x+3) = x+6 ………… 59/ x x x x x x x x + − + − = − + − 2 2 2 5 5 5 5 2 10 2 50 x x x x(x ) x(x ) (x ) + − + ⇔ − = − + − 2 5 5 5 5 2 5 2 25 x x x x(x ) x(x ) (x )(x ) + − + ⇔ − = − + − + 5 5 5 5 2 5 2 5 5 (1) * MTC : 2x(x–5)(x+5) * ĐKXĐ : x ≠ 0 ; x ≠ 5 ; x ≠ –5 * QĐ : (1) ⇔ 2(x+5) 2 – (x–5) 2 = x(x+5) ⇔ 2( x 2 +10x+25)–( x 2 –10x+25) = x 2 +5x ⇔ 2 x 2 +20x+50– x 2 +10x –25 – x 2 –5x = 0 ⇔ 25x = –25 ⇔ x = –1 (nhận) Vậy S= { } −1 60/ x x x x+ + + + + = + 1 2 3 4 99 98 97 96 x x x x ( ) ( ) ( ) ( ) + + + + ⇔ + + + = + + + 1 2 3 4 1 1 1 1 99 98 97 96 x x x x+ + + + + + + + ⇔ + = + 1 99 2 98 3 97 4 96 99 98 97 96 x x x x+ + + + ⇔ + = + 100 100 100 100 99 98 97 96 x x x x+ + + + ⇔ + − − = 100 100 100 100 0 99 98 97 96 (x )   ⇔ + + − − =  ÷   1 1 1 1 100 0 99 98 97 96 x⇔ + =100 0 x⇔ = −100 Vậy S= { } −100 61/ x x x x− − − − + + + + = 1909 1907 1905 1903 4 0 91 93 95 97 x x x ( ) ( ) ( ) − − − ⇔ + + + + + 1909 1907 1905 1 1 1 91 93 95 x ( ) − + + + =− 1903 1 4 0 97 4 x x x x− − − − ⇔ + + + = 2000 2000 2000 2000 0 91 93 95 97 ( x)   ⇔ − + + + =  ÷   1 1 1 1 2000 0 91 93 95 97 x⇔ − =2000 0 x⇔ = 2000 Vậy S= { } 2000 62/ x x x x− − − − + = + 5 15 1980 1990 1990 1980 15 5 x x x x ( ) ( ) ( ) ( ) − − − − ⇔ − + − = − + − 5 15 1980 1990 1 1 1 1 1990 1980 15 5 x x x x − − − − − − − − ⇔ + = + 5 1990 15 1980 1980 15 1990 5 1990 1980 15 5 x x x x− − − − ⇔ + = + 1995 1995 1995 1995 1990 1980 15 5 (x )   ⇔ − + − − =  ÷   1 1 1 1 1995 0 1990 1980 15 5 x⇔ − =1995 0 x⇔ = 1995 Vậy S= { } 1995 . TRƯỜNG THCS PHÚ THỌ QUẬN 11 GV: CAO MINH TÀI TRANG 1 CÁC BÀI TẬP MẪU 1/ x(x–1)(2–3x) = 0 ⇔ x = 0 hay x– 1 = 0 hay 2–3x = 0 ⇔ x= 0 hay x. (x–2)(x+2) = 0 ⇔ x–2 = 0 hay x+2 = 0 ⇔ x = 2 hay x = –2 21/ *Cách 1: 16 x 2 = 1 ⇔ x 2 = 1 16 ⇔ x = ± 1 4 *Cách 2: 16 x 2 = 1 ⇔ 16 x 2 –1 = 0 ⇔ (4x) 2 –1 2 = 0