Solve x=sqrt{2x-6}

$$x = \sqrt { 2 x - 6 }$$

Solve for x (complex solution)

$$x=1+\sqrt{5}i\approx 1+2.236067977i$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{2x-6}\right)^{2}$$

Calculate \sqrt{2x-6} to the power of 2 and get 2x-6.

$$x^{2}=2x-6$$

Subtract 2x from both sides.

$$x^{2}-2x=-6$$

Add 6 to both sides.

$$x^{2}-2x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 6}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-2\right)±\sqrt{4-24}}{2}$$

Add 4 to -24.

$$x=\frac{-\left(-2\right)±\sqrt{-20}}{2}$$

Take the square root of -20.

$$x=\frac{-\left(-2\right)±2\sqrt{5}i}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{5}i}{2}$$

Now solve the equation x=\frac{2±2\sqrt{5}i}{2} when ± is plus. Add 2 to 2i\sqrt{5}.

$$x=\frac{2+2\sqrt{5}i}{2}$$

Divide 2+2i\sqrt{5} by 2.

$$x=1+\sqrt{5}i$$

Now solve the equation x=\frac{2±2\sqrt{5}i}{2} when ± is minus. Subtract 2i\sqrt{5} from 2.

$$x=\frac{-2\sqrt{5}i+2}{2}$$

Divide 2-2i\sqrt{5} by 2.

$$x=-\sqrt{5}i+1$$

The equation is now solved.

$$x=1+\sqrt{5}i x=-\sqrt{5}i+1$$

Substitute 1+\sqrt{5}i for x in the equation x=\sqrt{2x-6}.

$$1+\sqrt{5}i=\sqrt{2\left(1+\sqrt{5}i\right)-6}$$

Simplify. The value x=1+\sqrt{5}i satisfies the equation.

$$1+i\times 5^{\frac{1}{2}}=1+i\times 5^{\frac{1}{2}}$$

Substitute -\sqrt{5}i+1 for x in the equation x=\sqrt{2x-6}.

$$-\sqrt{5}i+1=\sqrt{2\left(-\sqrt{5}i+1\right)-6}$$

Simplify. The value x=-\sqrt{5}i+1 does not satisfy the equation.

$$-i\times 5^{\frac{1}{2}}+1=-\left(1-i\times 5^{\frac{1}{2}}\right)$$

Equation x=\sqrt{2x-6} has a unique solution.

$$x=1+\sqrt{5}i$$