Algebra Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Expand $(1 + 2 \sqrt{x}) (1 + 2 \sqrt{x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $1$ by $1$ .

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

Step 2.3.1.3.1.2

Multiply $2 \sqrt{x}$ by $1$ .

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

Step 2.3.1.3.1.3

Multiply $2$ by $1$ .

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

Step 2.3.1.3.1.4

Multiply $2 \sqrt{x} (2 \sqrt{x})$ .

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Step 2.3.1.3.1.4.1

Multiply $2$ by $2$ .

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 \sqrt{x} \sqrt{x}$

Step 2.3.1.3.1.4.2

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

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

Step 2.3.1.3.1.4.3

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

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

Step 2.3.1.3.1.4.4

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

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 {\sqrt{x}}^{1 + 1}$

Step 2.3.1.3.1.4.5

Add $1$ and $1$ .

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 {\sqrt{x}}^{2}$

Step 2.3.1.3.1.5

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

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Step 2.3.1.3.1.5.1

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

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

Step 2.3.1.3.1.5.2

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

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

Step 2.3.1.3.1.5.3

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

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 x^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 x^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 x^{1}$

Step 2.3.1.3.1.5.5

Simplify.

$2 x + 1 = 1 + 2 \sqrt{x} + 2 \sqrt{x} + 4 x$

Step 2.3.1.3.2

Add $2 \sqrt{x}$ and $2 \sqrt{x}$ .

$2 x + 1 = 1 + 4 \sqrt{x} + 4 x$

Step 3

Solve for $4 \sqrt{x}$ .

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Step 3.1

Rewrite the equation as $1 + 4 \sqrt{x} + 4 x = 2 x + 1$ .

$1 + 4 \sqrt{x} + 4 x = 2 x + 1$

Step 3.2

Move all terms not containing $4 \sqrt{x}$ to the right side of the equation.

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$4 \sqrt{x} + 4 x = 2 x + 1 - 1$

Step 3.2.2

Subtract $4 x$ from both sides of the equation.

$4 \sqrt{x} = 2 x + 1 - 1 - 4 x$

Step 3.2.3

Combine the opposite terms in $2 x + 1 - 1 - 4 x$ .

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Step 3.2.3.1

Subtract $1$ from $1$ .

$4 \sqrt{x} = 2 x + 0 - 4 x$

Step 3.2.3.2

Add $2 x$ and $0$ .

$4 \sqrt{x} = 2 x - 4 x$

Step 3.2.4

Subtract $4 x$ from $2 x$ .

$4 \sqrt{x} = - 2 x$

Step 4

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

${(4 \sqrt{x})}^{2} = {(- 2 x)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Apply the product rule to $4 x^{\frac{1}{2}}$ .

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

Step 5.2.1.2

Raise $4$ to the power of $2$ .

$16 {(x^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$16 x^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$16 x^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$16 x^{1} = {(- 2 x)}^{2}$

Step 5.2.1.4

Simplify.

$16 x = {(- 2 x)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(- 2 x)}^{2}$ .

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Step 5.3.1.1

Apply the product rule to $- 2 x$ .

$16 x = {(- 2)}^{2} x^{2}$

Step 5.3.1.2

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

$16 x = 4 x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

Subtract $4 x^{2}$ from both sides of the equation.

$16 x - 4 x^{2} = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

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

$16 u - 4 u^{2} = 0$

Step 6.2.2

Factor $4 u$ out of $16 u - 4 u^{2}$ .

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Step 6.2.2.1

Factor $4 u$ out of $16 u$ .

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

Step 6.2.2.2

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

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

Step 6.2.2.3

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

$4 u (4 - u) = 0$

Step 6.2.3

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

$4 x (4 - x) = 0$

Step 6.3

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

$x = 0$

$4 - x = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

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

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Step 6.5.1

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

$4 - x = 0$

Step 6.5.2

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

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Step 6.5.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 6.5.2.2

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

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Step 6.5.2.2.1

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

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

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 6.6

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

$x = 0, 4$

Step 7

Exclude the solutions that do not make $\sqrt{2 x + 1} = 1 + 2 \sqrt{x}$ true.

$x = 0$