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

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

Solve for x (complex solution)

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

Solution Steps

Square both sides of the equation.

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

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

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

Expand \left(2\sqrt{1+x}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

$$x-1=4\left(1+x\right)$$

Use the distributive property to multiply 4 by 1+x.

$$x-1=4+4x$$

Subtract 4x from both sides.

$$x-1-4x=4$$

Combine x and -4x to get -3x.

$$-3x-1=4$$

Add 1 to both sides.

$$-3x=4+1$$

Add 4 and 1 to get 5.

$$-3x=5$$

Divide both sides by -3.

$$x=\frac{5}{-3}$$

Fraction \frac{5}{-3} can be rewritten as -\frac{5}{3} by extracting the negative sign.

$$x=-\frac{5}{3}$$

Substitute -\frac{5}{3} for x in the equation \sqrt{x-1}=2\sqrt{1+x}.

$$\sqrt{-\frac{5}{3}-1}=2\sqrt{1-\frac{5}{3}}$$

Simplify. The value x=-\frac{5}{3} satisfies the equation.

$$\frac{2}{3}i\times 6^{\frac{1}{2}}=\frac{2}{3}i\times 6^{\frac{1}{2}}$$

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

$$x=-\frac{5}{3}$$