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\left(\sqrt{2x-1}-1\right)^{2}=\left(\sqrt{x-1}\right)^{2}
Square both sides of the equation.
\left(\sqrt{2x-1}\right)^{2}-2\sqrt{2x-1}+1=\left(\sqrt{x-1}\right)^{2}
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{2x-1}-1\right)^{2}.
2x-1-2\sqrt{2x-1}+1=\left(\sqrt{x-1}\right)^{2}
Calculate \sqrt{2x-1} to the power of 2 and get 2x-1.
2x-2\sqrt{2x-1}=\left(\sqrt{x-1}\right)^{2}
Add -1 and 1 to get 0.
2x-2\sqrt{2x-1}=x-1
Calculate \sqrt{x-1} to the power of 2 and get x-1.
-2\sqrt{2x-1}=x-1-2x
Subtract 2x from both sides of the equation.
-2\sqrt{2x-1}=-x-1
Combine x and -2x to get -x.
\left(-2\sqrt{2x-1}\right)^{2}=\left(-x-1\right)^{2}
Square both sides of the equation.
\left(-2\right)^{2}\left(\sqrt{2x-1}\right)^{2}=\left(-x-1\right)^{2}
Expand \left(-2\sqrt{2x-1}\right)^{2}.
4\left(\sqrt{2x-1}\right)^{2}=\left(-x-1\right)^{2}
Calculate -2 to the power of 2 and get 4.
4\left(2x-1\right)=\left(-x-1\right)^{2}
Calculate \sqrt{2x-1} to the power of 2 and get 2x-1.
8x-4=\left(-x-1\right)^{2}
Use the distributive property to multiply 4 by 2x-1.
8x-4=x^{2}+2x+1
Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-1\right)^{2}.
8x-4-x^{2}=2x+1
Subtract x^{2} from both sides.
8x-4-x^{2}-2x=1
Subtract 2x from both sides.
6x-4-x^{2}=1
Combine 8x and -2x to get 6x.
6x-4-x^{2}-1=0
Subtract 1 from both sides.
6x-5-x^{2}=0
Subtract 1 from -4 to get -5.
-x^{2}+6x-5=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=6 ab=-\left(-5\right)=5
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=5 b=1
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.
\left(-x^{2}+5x\right)+\left(x-5\right)
Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).
-x\left(x-5\right)+x-5
Factor out -x in -x^{2}+5x.
\left(x-5\right)\left(-x+1\right)
Factor out common term x-5 by using distributive property.
x=5 x=1
To find equation solutions, solve x-5=0 and -x+1=0.
\sqrt{2\times 5-1}-1=\sqrt{5-1}
Substitute 5 for x in the equation \sqrt{2x-1}-1=\sqrt{x-1}.
2=2
Simplify. The value x=5 satisfies the equation.
\sqrt{2\times 1-1}-1=\sqrt{1-1}
Substitute 1 for x in the equation \sqrt{2x-1}-1=\sqrt{x-1}.
0=0
Simplify. The value x=1 satisfies the equation.
x=5 x=1
List all solutions of \sqrt{2x-1}-1=\sqrt{x-1}.