Solve 2-sqrt{x+1}=sqrt{2+x}

$$2 - \sqrt { x + 1 } = \sqrt { 2 + x }$$

Solve for x

$$x=-\frac{7}{16}=-0.4375$$

Solution Steps

Square both sides of the equation.

$$\left(2-\sqrt{x+1}\right)^{2}=\left(\sqrt{2+x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-\sqrt{x+1}\right)^{2}.

$$4-4\sqrt{x+1}+\left(\sqrt{x+1}\right)^{2}=\left(\sqrt{2+x}\right)^{2}$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$4-4\sqrt{x+1}+x+1=\left(\sqrt{2+x}\right)^{2}$$

Add 4 and 1 to get 5.

$$5-4\sqrt{x+1}+x=\left(\sqrt{2+x}\right)^{2}$$

Calculate \sqrt{2+x} to the power of 2 and get 2+x.

$$5-4\sqrt{x+1}+x=2+x$$

Subtract x from both sides.

$$5-4\sqrt{x+1}+x-x=2$$

Combine x and -x to get 0.

$$5-4\sqrt{x+1}=2$$

Subtract 5 from both sides.

$$-4\sqrt{x+1}=2-5$$

Subtract 5 from 2 to get -3.

$$-4\sqrt{x+1}=-3$$

Divide both sides by -4.

$$\sqrt{x+1}=\frac{-3}{-4}$$

Fraction \frac{-3}{-4} can be simplified to \frac{3}{4} by removing the negative sign from both the numerator and the denominator.

$$\sqrt{x+1}=\frac{3}{4}$$

Square both sides of the equation.

$$x+1=\frac{9}{16}$$

Subtract 1 from both sides of the equation.

$$x+1-1=\frac{9}{16}-1$$

Subtracting 1 from itself leaves 0.

$$x=\frac{9}{16}-1$$

Subtract 1 from \frac{9}{16}.

$$x=-\frac{7}{16}$$

Substitute -\frac{7}{16} for x in the equation 2-\sqrt{x+1}=\sqrt{2+x}.

$$2-\sqrt{-\frac{7}{16}+1}=\sqrt{2-\frac{7}{16}}$$

Simplify. The value x=-\frac{7}{16} satisfies the equation.

$$\frac{5}{4}=\frac{5}{4}$$

Equation -\sqrt{x+1}+2=\sqrt{x+2} has a unique solution.

$$x=-\frac{7}{16}$$