Solve x=sqrt{2x-9}

$$x = \sqrt { 2 x - 9 }$$

Solve for x (complex solution)

$$x=1+2\sqrt{2}i\approx 1+2.828427125i$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{2x-9}\right)^{2}$$

Calculate \sqrt{2x-9} to the power of 2 and get 2x-9.

$$x^{2}=2x-9$$

Subtract 2x from both sides.

$$x^{2}-2x=-9$$

Add 9 to both sides.

$$x^{2}-2x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 9}}{2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-\left(-2\right)±\sqrt{4-36}}{2}$$

Add 4 to -36.

$$x=\frac{-\left(-2\right)±\sqrt{-32}}{2}$$

Take the square root of -32.

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

The opposite of -2 is 2.

$$x=\frac{2±4\sqrt{2}i}{2}$$

Now solve the equation x=\frac{2±4\sqrt{2}i}{2} when ± is plus. Add 2 to 4i\sqrt{2}.

$$x=\frac{2+4\sqrt{2}i}{2}$$

Divide 2+4i\sqrt{2} by 2.

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

Now solve the equation x=\frac{2±4\sqrt{2}i}{2} when ± is minus. Subtract 4i\sqrt{2} from 2.

$$x=\frac{-4\sqrt{2}i+2}{2}$$

Divide 2-4i\sqrt{2} by 2.

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

The equation is now solved.

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

Substitute 1+2\sqrt{2}i for x in the equation x=\sqrt{2x-9}.

$$1+2\sqrt{2}i=\sqrt{2\left(1+2\sqrt{2}i\right)-9}$$

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

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

Substitute -2\sqrt{2}i+1 for x in the equation x=\sqrt{2x-9}.

$$-2\sqrt{2}i+1=\sqrt{2\left(-2\sqrt{2}i+1\right)-9}$$

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

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

Equation x=\sqrt{2x-9} has a unique solution.

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