Solve x=sqrt{2x-1}

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

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

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

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

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

Subtract 2x from both sides.

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

Add 1 to both sides.

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

To solve the equation, factor x^{2}-2x+1 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=1$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-1\right)\left(x-1\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x-1=0.

$$x=1$$

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

$$1=\sqrt{2\times 1-1}$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

Equation x=\sqrt{2x-1} has a unique solution.

$$x=1$$