Solve (x)=sqrt{x-1}

$$( x ) = \sqrt { x - 1 }$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{3}i}{2}\approx 0.5+0.866025404i$$

Solution Steps

Square both sides of the equation.

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

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

$$x^{2}=x-1$$

Subtract x from both sides.

$$x^{2}-x=-1$$

Add 1 to both sides.

$$x^{2}-x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4}}{2}$$

Add 1 to -4.

$$x=\frac{-\left(-1\right)±\sqrt{-3}}{2}$$

Take the square root of -3.

$$x=\frac{-\left(-1\right)±\sqrt{3}i}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{3}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{3}i}{2} when ± is plus. Add 1 to i\sqrt{3}.

$$x=\frac{1+\sqrt{3}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{3}i}{2} when ± is minus. Subtract i\sqrt{3} from 1.

$$x=\frac{-\sqrt{3}i+1}{2}$$

The equation is now solved.

$$x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2}$$

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

$$\frac{1+\sqrt{3}i}{2}=\sqrt{\frac{1+\sqrt{3}i}{2}-1}$$

Simplify. The value x=\frac{1+\sqrt{3}i}{2} satisfies the equation.

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

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

$$\frac{-\sqrt{3}i+1}{2}=\sqrt{\frac{-\sqrt{3}i+1}{2}-1}$$

Simplify. The value x=\frac{-\sqrt{3}i+1}{2} does not satisfy the equation.

$$-\frac{1}{2}i\times 3^{\frac{1}{2}}+\frac{1}{2}=-\left(\frac{1}{2}-\frac{1}{2}i\times 3^{\frac{1}{2}}\right)$$

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

$$x=\frac{1+\sqrt{3}i}{2}$$