The value of x is [tex] \frac{1}{4} [/tex]
Step-by-step explanation:
Given:
[tex] \sqrt{x} + \sqrt{x + 2} = 2[/tex]
Rearranging the radical on left side,
[tex]\sqrt{x + 2} = 2 - \sqrt{x}[/tex]
Power on both sides,
[tex](\sqrt{x + 2})^{2} = (2 - \sqrt{x})^{2} [/tex]
Simplifying the left,
[tex]x + 2 = (2 - \sqrt{x} )^{2}[/tex]
For the RHS equation, use the property of (a-b)² = (a²-2ab+b²),
[tex] = > x + 2 = {2}^{2} - (2 \times 2 \sqrt{x}) + ( \sqrt{x})^{2} [/tex]
Now calculating its powers,
[tex] = > x + 2 = 4 - 4\sqrt{x} + x[/tex]
Now sending -4√x to the LHS (left side), its sign becomes plus (+),
[tex] = > 4 \sqrt{x} = 4 + x - x - 2[/tex]
Now the +x and -x will be cancelled,
[tex] = > 4 \sqrt{x} = 4 - 2[/tex]
[tex] = > 4 \sqrt{x} = 2[/tex]
Bringing 4 to the right side, it becomes the denominator,
[tex] = > \sqrt{x} = \frac{2}{4} [/tex]
[tex] = > \sqrt{x} = \frac{1}{4} [/tex]
Now powering both sides,
[tex] = > ( \sqrt{x})^{2} = ( \frac{1}{2})^{2}[/tex]
[tex] = > x = \frac{1}{4} [/tex]
