Chuyên đề Tìm X - Bồi dưỡng học sinh giỏi Toán 7

 CHUYÊN ĐỀ TÌM X 
 DẠNG 1: TÌM X THÔNG THƯỜNG 
Bài 1: Tìm x biết: 
 xx 3 13 7 7 13 5 6
a, − − = + x b, +=
 2 5 5 5 10 x−1 2 x − 2 3 x − 3 
HD: 
 x 313 x 77 x 31377 x x 37713 x x
 a, − − =+ xx = −+=+ = −−=− 
 2 55 510 255510 251055
 −4 − 6 − 6 − 4 3 3
 => xx= = =: = . Vậy x = 
 5 5 5 5 2 2
 13 5 6 78 15 12 93 12
 b, + = = + = = = = =93 12 (Vô lý) 
 x−12233 x − x − 666666 x − x − x − 6666 x − x −
Bài 2: Tìm x biết: 
 2xx− 3 − 3 5 − 3 1 2 3 4 7
a, + = − b, − = − − 2
 3 2 6 3 3xx 12 5 
HD: 
 233531x− − − x4x − 6 +( − 9) 532 − x − 18
 a, += −= = = −=−= == =4x 15 3 3 x 7 x 18 x 
 3 2 6 3 6 6 7
 2347 2147 2741 2361
 b, − =− −2 = −=−+= 2 +=++= 2 = 
 3125x x 345 x x 3 x x 54 320 x
 460 460
 = 3xx = = = 
 61 183
Bài 3: Tìm x biết: 
 1− 2 3 6 5 9 13 5
a, + − = b, x+15 x − 20 x = 16
 xx−−1 3 4 5 2 2 17 17 17 
HD: 
 1 3 5 1 5−− 3 7 3
 a, + = = + = = = 
 x−11022 − x x − 12( x − 110) 2( x − 110)
 70−− 35 32
 = 2( x −=− 1) = −= x 1 = = x 
 3 3 3
 9 13 5
 b, => +15 − 20x == −== =− 16 4. x 16 x 4 
 17 17 17
Bài 4: Tìm x biết: 
 3
a, 720: 41−( 2x − 5) = 2 .5 b, 6(xx+ 11) − 7( 2 −) = 26
HD: 
 23 23
 a, => 720 : 46− 2x == −== = 40 46 2 x 18 2 x 46 :18 = = = x 
 9 18
 b, => 6x+ 66 − 14 + 7 x = 26 = 13 x =− 26 = =− x 2 
GV: Ngô Thế Hoàng_THCS Hợp Đức 1 
 DẠNG 2: ĐƯA VỀ TÍCH BẰNG 0 
Bài 1: Tìm x biết: 
 x−3 x − 3 x − 3 x − 3 x+1 x + 3 x + 5 x + 7
a, + = + b, +=+ 
 13 14 15 16 65 63 61 59
HD: 
 x−3 x − 3 x − 3 x − 3 1 1 1 1
 a, = + − − == −0(x 3) +−−= 0 
 13 14 15 16 13 14 15 16
 11 11 1 1 1 1
 => x = 3 vì − 0 và − 0 nên + − − 0 
 13 15 14 16 13 14 15 16
 x+1 x + 3 x + 5 x + 7 x + 66 x + 66 x + 66 x + 66
 b, = ++1 += 1 ++ 1 += 1 + = + 
 65 63 61 59 65 63 61 59
 1 1 1 1 1 1 1 1
 => (xx+66) + − − = 0 = = − 61 vì + − − 0 
 65 63 61 59 65 63 61 59
Bài 2: Tìm x, biết: 
 29−x 27 − x 25 − x 23 − x 21 − x x−10 x − 14 x − 5 x − 148
a, + + + + = −5 b, + + + = 0 
 21 23 25 27 29 30 43 95 8
HD: 
 29−x 27 − x 25 − x 23 − x 21 − x 
 a, = ++1 ++ 1 ++ 1 ++ 1 += 1 0 
 21 23 25 27 29 
 50−−−−−xxxxx 50 50 50 50 1 1 1 1 1
 => + + + + = 0 => (50−x) + + + + = 0 
 21 23 25 27 29 21 23 25 27 29
 x−10 x − 14 x − 5 x − 148 
 b, => −3 + − 2 + − 1 + + 6 = 0 
 30 43 95 8 
 xxxx−−−−100 100 100 100 1 1 1 1
 => + + + = 0 =>(x −100) + + + = 0 
 30 43 95 8 30 43 95 8
Bài 3, Tìm x, biết: 
 x−5 x − 4 x − 3 x − 100 x − 101 x − 102 x−2 x − 1 x − 4 x − 3
a, + + = + + b, + = + 
 100 101 102 5 4 3 7 8 5 6
HD: 
 x−5 x − 4 x − 3 x − 100 x − 101 x − 102 
 a, = −+1 −+ 1 −= 1 −+ 1 −+ 1 − 1 
 100 101 102 5 4 3 
 x−105 x − 105 x − 105 x − 105 x − 105 x − 105
 => + + = + + 
 100 101 102 5 4 3
 => xx−105 = 0 = = 105 
 x−2 x − 1 x − 4 x − 3 x − 9 x − 9 x − 9 x − 9
 b, => −+1 −= 1 −+ 1 −= 1 + = + 
 7 8 5 6 7 8 5 6
 => xx−9 = 0 = = 9 
Bài 4, Tìm x, biết: 
 x+1 x + 2 x + 3 x + 4 x + 5 x + 6 2x+ 19 2 x + 17 2 x + 7 2 x + 5
a, + + = + + b, − = − 
 94 93 92 91 90 89 21 23 33 35
HD: 
 x+1 x + 2 x + 3 x + 4 x + 5 x + 6 
 a, => ++1 ++ 1 += 1 ++ 1 ++ 1 + 1 
 94 93 94 91 90 89 
GV: Ngô Thế Hoàng_THCS Hợp Đức 3 
Bài 8: Tìm x, biết: 
 x−1 x − 2 x − 3 x − 4 59−x 57 − x 55 − x 53 − x 51 − x
a, + = + b, + + + + = −5 
 2009 2008 2007 2006 41 43 45 47 49
HD: 
 x−1 x − 2 x − 3 x − 4 
 a, = −1 + − 1 = − 1 + − 1 
 2009 2008 2007 2006 
 x−2010 x − 2010 x − 2010 x − 2010
 = + = + = xx −2010 = 0 = = 2010 
 2009 2008 2007 2006
 59−x 57 − x 55 − x 53 − x 51 − x 
 b, = ++1 ++ 1 ++ 1 ++ 1 += 1 0 
 41 43 45 47 49 
 100−−−−−xxxxx 100 100 100 100
 = + + + + == −== =0 100xx 0 100 
 41 43 45 47 49
Bài 9: Tìm x, biết: 
 x+14 x + 15 x + 16 x + 17 x−90 x − 76 x − 58 x − 36 x − 15
a, + + + = −4 b, + + + + =15 
 86 85 84 83 10 12 14 16 17
HD: 
 x+14 x + 15 x + 16 x + 17 
 a, = +1 + + 1 + + 1 + + 1 = 0 
 86 85 84 83 
 xxxx++++100 100 100 100
 = + + + == +== =−0xx 100 0 100 
 86 85 84 83
 x−90 x − 76 x − 58 x − 36 x − 15 
 b, => −1 + − 2 + − 3 + − 4 + − 5 = 0 
 10 12 14 16 17 
 xxxxx−−−−−100 100 100 100 100
 = + + + + == −== =0xx 100 0 100 
 10 12 14 16 17
Bài 10, Tìm x, biết: 
 x−1 x − 2 x − 3 x − 4 x+1 x + 1 x + 1 x + 1
a, + − = b, + = + 
 2011 2010 2009 2008 11 12 13 14
HD: 
 x−1 x − 2 x − 3 x − 4 
 a, = −1 + − 1 − − 1 − − 1 = 0 
 2011 2010 2009 2008 
 x−2012 x − 2012 x − 2012 x − 2012
 => + − − == −0xx 2012 == = 0 2012 
 2011 2010 2009 2008
 1 1 1 1
 b, = (x +1) + − − == +== =− 0 x 1 0 x 1 
 11 12 13 14
 x+1 x + 1 x + 1 x + 1 x + 1
Bài 11, Tìm x, biết: + + = + 
 10 11 12 13 14
HD: 
 1 1 1 1 1
 = (x +1) + + − − == +== =− 0 x 1 0 x 1 
 10 11 12 13 14
Bài 12, Tìm x, biết: 
 2 4 6 x
a, + + = 
 (x+2)( x + 4) ( x + 4)( x + 8) ( x + 8)( x + 14) ( x + 2)( x + 14)
HD: 
GV: Ngô Thế Hoàng_THCS Hợp Đức 5 
 DẠNG 3: SỬ DỤNG TÍNH CHẤT LŨY THỪA 
Bài 1: Tìm x biết: 
 2 10
 11 x 2 22
a, 30 xx+ − + = b, xx+12 = + c, ( x +=3) 1024.125 .25 
 2 3 2 
Bài 2: Tìm x biết: 
 x
 64 4.59 .7 10− 5 11 .7 9
a, 9( 5x− 1) − 16( 5 x − 1) + 36( 5 x − 1) = 15 b, = 9 
 27 35 .4
Bài 3: Tìm x biết: 
 5
a, 16(xx− 1) − 9 − 9 = 5 b, (x −1) = − 243 c, 3−2 .3 4 .3n = 3 7 
Bài 4: Tìm x biết: 
a, (3xx− 1)10 =( 3 − 1) 20 b, x(66− x)2003 =( − x) 2003 c, 5xx+= 5+2 650 
HD: 
 1
 3x −= 1 0 x =
 20 10 10 10 
 a, = −−−== −(31x) ( 310313110 x) ( x) ( x −−== ) 10 = 3 
 (3x −= 1) 1 
 3x − 1 = 1
 2003 2003 2003 6−xx = 0 = 6
 b, => x(6− x) −−( 6 x) == − 0( 6 x) ( x −== 1) 0 = 
 xx−1 = 0 = 1
 c, => 5x+ 5 x .52 = 650 = 5 x( 1 + 25) = 650 = 5 x = 25 
Bài 5: Tìm x biết: 
a, 5xx+= 5+2 650 b, 325 .3n = 3 c, (22 : 4) .2n = 4 
Bài 6: Tìm x biết: 
 1 n n n nn 1 nn−1
a, .2+= 4.2 9.5 b, 2 += 4 9.5 c, 2 .9= 9.5 
 2 2
Bài 7: Tìm x biết: 
a, 2nn+3 .2= 144 b, xx−2015xx++2 − − 2015 12 = 0 c, 2xx− 186 = 2 − 1 
 ( ) ( ) ( ) ( )
 7
Bài 8: Tìm x biết: 2xx−−12+= 5.2 
 32
HD: 
 x−1 5 7 x − 1 7 7 x − 1 1 − 4
 = 2 1 + = = 2 . = = 2 = = 2 = x = − 3 
 2 32 2 32 16
Bài 9: Tìm x biết: 
a, (xx−7)xx++1 −( − 7) 11 = 0 b, 2x++21 .3 x .5 x = 10800 c, (2xx− 15)53 =( 2 − 15) 
HD: 
 x+1 10 x −=70 x = 7
 a, xx−7 1 − − 7 = 0 = = 
 ( ) ( ) 10 
 (x −=71) x −71 = 
 10800 x
 b, 2x .4.3 x .3.5 x= 10800 = 2 x .3 x .5 x = == 900( 2.3.5) == === = 900 30 x 900 302 x 2 
 12
GV: Ngô Thế Hoàng_THCS Hợp Đức 7