Solve by Factoring square root of 2x-1=x

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

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{2 x - 1} - x = 0$

Step 2

Add $x$ to both sides of the equation.

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

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 1}$ as ${(2 x - 1)}^{\frac{1}{2}}$ .

${({(2 x - 1)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(2 x - 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x - 1)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

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

Simplify.

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

Step 5

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

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

Factor the left side of the equation.

Factor $- 1$ out of $2 x - 1 - x^{2}$ .

Reorder the expression.

Move $- 1$ .

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

Reorder $2 x$ and $- x^{2}$ .

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

Factor $- 1$ out of $- x^{2}$ .

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

Factor $- 1$ out of $2 x$ .

$- (x^{2}) - (- 2 x) - 1 = 0$

Rewrite $- 1$ as $- 1 (1)$ .

$- (x^{2}) - (- 2 x) - 1 \cdot 1 = 0$

Factor $- 1$ out of $- (x^{2}) - (- 2 x)$ .

$- (x^{2} - 2 x) - 1 \cdot 1 = 0$

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (1)$ .

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

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 1 + 1^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

$- {(x - 1)}^{2} = 0$

Divide each term in $- {(x - 1)}^{2} = 0$ by $- 1$ and simplify.

Divide each term in $- {(x - 1)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 1)}^{2}}{- 1} = \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{{(x - 1)}^{2}}{1} = \frac{0}{- 1}$

Divide ${(x - 1)}^{2}$ by $1$ .

${(x - 1)}^{2} = \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

${(x - 1)}^{2} = 0$

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Solve by Factoring square root of 2x-1=x

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