Answer:
x = -1
Step-by-step explanation:
[tex]2 + \sqrt{x+1} - \sqrt{x+5} = 0[/tex]
Subtract 2 to both sides
[tex]2 + \sqrt{x+1} - \sqrt{x+5} - 2 = 0 -2[/tex]
[tex]\sqrt{x+1} - \sqrt{x+5} = -2[/tex]
Raise both sides by 2
[tex](\sqrt{x+1} -\sqrt{x+5} )^2 = -2^2[/tex]
[tex]-\sqrt{(x+5)(x+1)} - \sqrt{(x+5)(x+1)} + 2x + 6 = 4[/tex]
[tex]-2\sqrt{(x+1)(x+5)} + 2x + 6 = 4[/tex]
Subtract 6 to both sides
[tex]-2\sqrt{(x+1)(x+5)} + 2x + 6-6 = 4-6[/tex]
[tex]-2\sqrt{(x+1)(x+5)} + 2x = -2[/tex]
Divide both sides by -2
[tex]\sqrt{(x+1)(x+5)} -x = 1[/tex]
Raise both sides by 2
[tex](\sqrt{(x+1)(x+5)} -x)^2=1^2[/tex]
[tex](\sqrt{(x+1)(x+5)}^2+x^2=1[/tex]
[tex]x^2+6x+5 + x^2=1[/tex]
[tex]2x^2 + 6x + 5 = 1[/tex]
[tex]2x^2 + 6x + 4 = 0[/tex]
Divide both sides by 2
[tex]x^2 + 3x + 2[/tex]
Factor
[tex](x+1)(x+2)[/tex]
x = -1, x = -2
Ignore -2 because if we substitute -2 to the equation, it doesn't satisfy the equation.
Therefore x = -1
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