Solve sqrt{x-1}=x

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

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.

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

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

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

Subtract x^{2} from both sides.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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}.

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times -1.

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

Add 1 to -4.

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

Take the square root of -3.

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

Multiply 2 times -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}.

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

Divide -1+i\sqrt{3} by -2.

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

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

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

Divide -1-i\sqrt{3} by -2.

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

The equation is now solved.

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

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

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

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

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

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

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

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

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

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

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