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[tex] \sqrt{x + 9} = x - 3[/tex]
Sides gets power 2
[tex] ({ \sqrt{x + 9} })^{2} = ( {x - 3})^{2} [/tex]
[tex] |x + 9| = {x}^{2} - 6x + 9 [/tex]
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[tex]x + 9 = {x}^{2} - 6x + 9[/tex]
Subtract sides 9
[tex]x + 9 - 9 = {x}^{2} - 6x + 9 - 9[/tex]
[tex]x = {x}^{2} - 6x[/tex]
Subtract sides x
[tex]x - x = {x}^{2} - 6x - x[/tex]
[tex] {x}^{2} - 7x = 0[/tex]
[tex]x(x - 7) = 0[/tex]
[tex]x = 0[/tex]
OR
[tex]x - 7 = 0[/tex]
[tex] x = 7[/tex]
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[tex]x + 9 = - ( {x}^{2} - 6x + 9)[/tex]
[tex]x + 9 = - {x}^{2} + 6x - 9[/tex]
Subtract sides 9
[tex]x + 9 - 9 = - {x}^{2} + 6x - 9 - 9 \\ [/tex]
[tex]x = - {x}^{2} + 6x - 18 [/tex]
Subtract sides x
[tex]x - x = - {x}^{2} + 6x - 18 - x [/tex]
[tex] - {x}^{2} + 5x - 18 = 0[/tex]
Multiply sides by -1
[tex] {x}^{2} - 5x + 18 = 0 [/tex]
[tex]x = No \: \: solution[/tex]
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CHECK the roots :
[tex]x = 0[/tex]
[tex] \sqrt{0 + 9} ≠0 - 3[/tex]
[tex] \sqrt{9} ≠ - 3[/tex]
[tex]3≠ - 3[/tex]
Thus (( x = 0 )) is unacceptable.
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[tex]x = 7[/tex]
[tex] \sqrt{7 + 9} = 7 - 3[/tex]
[tex] \sqrt{16} = 4[/tex]
[tex]4 = 4[/tex]
Thus (( x = 7 )) is the only solution for x.
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Done...
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