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Prove that the following equations has no solutions. (i) sqrt((2x+7))+sqrt((x+4))=0 (ii) sqrt((x-4))=-5 (iii) sqrt((6-x))-sqrt((x-8))=2 (iv) sqrt(-2-x)=root(5)((x-7)) (v) sqrt(x)+sqrt((x+16))=3 (vi) 7sqrt(x)+8sqrt(-x)+15/(x^(3))=98 (vii) sqrt((x-3))-sqrt(x+9)=sqrt((x-1)) |
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Answer» Solution :(i) We have `sqrt((2x+7))+sqrt((x+4))=0` This equation is defined for `2x+7ge0` and `x+4ge0implies{(xge-7/2),(xge-4):"` `:.xge-7/2` For `xge-7/2` , the left hand SIDE of the original equation is positive, but right hand side is ZERO. Therefore, the equation has no roots. (ii) We have `sqrt(x-4))=-5` The equation is defined for `x-4ge0` `:.xge4` For `xge4`, the left hand side of the original equation is positive but right hand side is negative. Therefore, the equation has no roots. (iii) We have `sqrt((5-x))-sqrt(x-8)=2` The equation is defined for `6-xge0` and `x-8ge0` `:.{(xle6),(xge8):}` Consequently, there is no `x` for which both expressions WOULD have sense. Therefore, the equation has no roots. (iv) We have `sqrt((-2-x))=root(5)((x-7))` This equation is defined for `-2-xge0impliesxle-2` For `xle-2` the left hand side is positive, but right hand side is negative. Therefore, the equation has no roots. (v) We have `sqrt(x)+sqrt((x+16))=3` The equation isdefined for `xge0` and `x+16ge0implies{(xge0),(xge-16):}` Hence `xge0` For `xge0` the left hand side `GE4`, but right hand side is 3. Therefore, the equation has no roots. (vi) We have `7sqrt(x)+8sqrt(-x)+15/(x^(3))=98` For `xlt0` the expression `7sqrt(x)` is meaningless, For `xgt0`, the expressionn `8sqrt(-x)` is meaningles and for `x=0` the expression `15/(x^(3))` is meaningless. Consequently the left hand side of the original equation ils meaningless for any `x epsilonR`. Therefore the equation has no roots. (vii) We have `sqrt((x-3))-sqrt((x+9))=sqrt(x-1)` This equation is defined for `{(x-3ge0),(x+9ge0),(x-1ge0):}implies{(xge3),(xge-9),(xge1):}` Hence `xge3` For `xge3, sqrt(x-3)ltsqrt(x+9)` i.e. `sqrt((x-3))-sqrt((x+9))LT0` Hence for `xge3`,the left hand side of the original equation is negative and right hand side is positive. Therefore, the equation has no roots. |
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