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

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Move all terms not containing $\sqrt{2 x + 1}$ to the right side of the equation.

Add $\sqrt{x}$ to both sides of the equation.

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

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Simplify ${(\sqrt{x} + 1)}^{2}$ .

Rewrite ${(\sqrt{x} + 1)}^{2}$ as $(\sqrt{x} + 1) (\sqrt{x} + 1)$ .

$2 x + 1 = (\sqrt{x} + 1) (\sqrt{x} + 1)$

Expand $(\sqrt{x} + 1) (\sqrt{x} + 1)$ using the FOIL Method.

Apply the distributive property.

$2 x + 1 = \sqrt{x} (\sqrt{x} + 1) + 1 (\sqrt{x} + 1)$

Apply the distributive property.

$2 x + 1 = \sqrt{x} \sqrt{x} + \sqrt{x} \cdot 1 + 1 (\sqrt{x} + 1)$

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x} \sqrt{x}$ .

Raise $\sqrt{x}$ to the power of $1$ .

$2 x + 1 = {\sqrt{x}}^{1} \sqrt{x} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Raise $\sqrt{x}$ to the power of $1$ .

$2 x + 1 = {\sqrt{x}}^{1} {\sqrt{x}}^{1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x + 1 = {\sqrt{x}}^{1 + 1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Add $1$ and $1$ .

$2 x + 1 = {\sqrt{x}}^{2} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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

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

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

$2 x + 1 = x^{\frac{1}{2} \cdot 2} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Combine $\frac{1}{2}$ and $2$ .

$2 x + 1 = x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 1 = x^{\frac{2}{2}} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Rewrite the expression.

$2 x + 1 = x^{1} + \sqrt{x} \cdot 1 + 1 \sqrt{x} + 1 \cdot 1$

Simplify.

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

Multiply $\sqrt{x}$ by $1$ .

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

Multiply $\sqrt{x}$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $\sqrt{x}$ and $\sqrt{x}$ .

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

Step 5

Solve for $2 \sqrt{x}$ .

Rewrite the equation as $x + 2 \sqrt{x} + 1 = 2 x + 1$ .

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

Move all terms not containing $2 \sqrt{x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

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

Subtract $1$ from both sides of the equation.

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

Combine the opposite terms in $2 x + 1 - x - 1$ .

Subtract $1$ from $1$ .

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

Add $2 x - x$ and $0$ .

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

Subtract $x$ from $2 x$ .

$2 \sqrt{x} = x$

Step 6

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

${(2 \sqrt{x})}^{2} = x^{2}$

Step 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $2 x^{\frac{1}{2}}$ .

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

Raise $2$ to the power of $2$ .

$4 {(x^{\frac{1}{2}})}^{2} = x^{2}$

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

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

$4 x^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

$4 x^{1} = x^{2}$

Simplify.

$4 x = x^{2}$

Step 8

Solve for $x$ .

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

$4 x - x^{2} = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u - u^{2} = 0$

Factor $u$ out of $4 u - u^{2}$ .

Factor $u$ out of $4 u$ .

$u \cdot 4 - u^{2} = 0$

Factor $u$ out of $- u^{2}$ .

$u \cdot 4 + u (- u) = 0$

Factor $u$ out of $u \cdot 4 + u (- u)$ .

$u (4 - u) = 0$

Replace all occurrences of $u$ with $x$ .

$x (4 - x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $4 - x$ equal to $0$ and solve for $x$ .

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Solve $4 - x = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

The final solution is all the values that make $x (4 - x) = 0$ true.

$x = 0, 4$

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

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