Solve sqrt{x-5}=2sqrt{x}

$$\sqrt{ x-5 } =2 \sqrt{ x }$$

Solve for x (complex solution)

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

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x-5} to the power of 2 and get x-5.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

$$x-5=4x$$

Subtract 4x from both sides.

$$x-5-4x=0$$

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

$$-3x-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-3x=5$$

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-5}=2\sqrt{x}.

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

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

$$\frac{2}{3}i\times 15^{\frac{1}{2}}=\frac{2}{3}i\times 15^{\frac{1}{2}}$$

Equation \sqrt{x-5}=2\sqrt{x} has a unique solution.

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