Solve 2sqrt{2x-1}-x=1

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

Solve for x

$$x=5$$

$$x=1$$

Solution Steps

Subtract -x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(2\sqrt{2x-1}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

$$4\left(2x-1\right)=\left(1+x\right)^{2}$$

Use the distributive property to multiply 4 by 2x-1.

$$8x-4=\left(1+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$8x-4=1+2x+x^{2}$$

Subtract 1 from both sides.

$$8x-4-1=2x+x^{2}$$

Subtract 1 from -4 to get -5.

$$8x-5=2x+x^{2}$$

Subtract 2x from both sides.

$$8x-5-2x=x^{2}$$

Combine 8x and -2x to get 6x.

$$6x-5=x^{2}$$

Subtract x^{2} from both sides.

$$6x-5-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+6x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

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

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

$$\left(-x^{2}+5x\right)+\left(x-5\right)$$

Factor out -x in -x^{2}+5x.

$$-x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+1\right)$$

To find equation solutions, solve x-5=0 and -x+1=0.

$$x=5 x=1$$

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

$$2\sqrt{2\times 5-1}-5=1$$

Simplify. The value x=5 satisfies the equation.

$$1=1$$

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

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

Simplify. The value x=1 satisfies the equation.

$$1=1$$

List all solutions of 2\sqrt{2x-1}=x+1.

$$x=5 x=1$$