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Resol les següents equacions: (MV) a) `sqrt(2x-3) + 1= x`
`sqrt(2x-3)^2 = (x - 1)^2` `2x - 3 = x^2 - 2x + 1` `0 = -2x + 3 + x^2 - 2x + 1` `0 = x^2 - 4x + 4` `x=(4\pmsqrt((-4)^2-4·1·4))/(2·1)` `x=(4\pmsqrt(16-16))/2` `x=(4\pmsqrt(0))/2` `x=(4\pm 0)/2` `x =2` b) `sqrt(x+6) - sqrt(x-2) = 2`
`(sqrt(x+6))^2 = (sqrt(x-2) + 2)^2` `x+6 = (sqrt(x-2))^2 + 2· sqrt(x-2) · 2 + 2^2` `x + 6 = x - 2 + 4sqrt(x-2) + 4` `x + 6 - x + 2 - 4 = 4sqrt(x-2)` `4 = 4sqrt(x-2)` `4/4 = sqrt(x-2)` `1 = sqrt(x-2)` `1^2 = (sqrt(x-2))^2` `1 = x - 2` `1 + 2 = x ` `x = 3` c) `sqrt(5x+4) - 1 = 2x`
`sqrt(5x+4)^2 = (1 + 2x)^2` `5x + 4 = 1^2 + 2·1·2x + (2x)^2` `0 = -5x - 4 + 1 + 4x + 4x^2` `0 = 4x^2 - x - 3` `x=(1\pmsqrt((-1)^2-4·4·(-3)))/(2·4)` `x=(1\pmsqrt(1+48))/8` `x=(1\pmsqrt(49))/8` `x=(1\pm 7)/8` `x_1 =(1+7)/8 = 1` `x_2 =(1-7)/8 = -6/8 = -3/4` d) `sqrt(x+19) + 1 = sqrt(2x+4)`
`sqrt(x+19)^2 = (sqrt(2x+4) - 1)^2` `x+19 = sqrt(2x+4)^2 - 2·sqrt(2x+4) + 1^2)` `x+19 = (2x+4) - 2sqrt(2x+4) + 1` `x+19 = 2x+4 - 2sqrt(2x+4) + 1` `x+19 - 2x -4 -1 = - 2sqrt(2x+4)` `-x + 14 = - 2sqrt(2x+4)` `x - 14 = 2sqrt(2x+4)` `(x - 14)^2 = (2sqrt(2x+4))^2` `x^2 - 28x + 14^2 = 4 · (2x+4)` `x^2 - 28x + 196 = 8x + 16` `x^2 - 28x + 196 -8x - 16 = 0` `x^2 - 36x + 180 = 0` `x=(36\pmsqrt((-36)^2-4·180))/(2·1)` `x=(36\pmsqrt(1296-720))/2` `x=(36\pmsqrt(576))/2` `x=(36\pm 24)/2` `x_1 =(36+24)/2 = 30` `x_2 =(36-24)/2 = 6` e) `3·sqrt(x-1) + 11 = 2x`
`(3sqrt(x-1))^2 = (2x - 11)^2` `9(x-1) = 4x^2 - 44x + 121` `9x - 9 = 4x^2 - 44x + 121` `0 = 4x^2 - 44x + 121 -9x +9` `0 = 4x^2 - 53x + 130` `4x^2 - 53x + 130 = 0` `x=(53\pmsqrt(53^2-4·4·130))/(2·4)` `x=(53\pmsqrt(729))/8` `x=(53\pm 27)/8` `x_1 =(53+27)/8 = 10` `x_2 =(53-27)/8 = 26/8 = 13/4` |