Algebra Examples

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

Step 1

Write $f (x) = x + 2 \sqrt{x - 1}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Subtract $y$ from both sides of the equation.

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

Step 3.3

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

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

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

Apply the product rule to $2 {(y - 1)}^{\frac{1}{2}}$ .

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

Step 3.4.2.1.2

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

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

Step 3.4.2.1.3

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

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Step 3.4.2.1.3.1

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

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

Step 3.4.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.3.2.1

Cancel the common factor.

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

Step 3.4.2.1.3.2.2

Rewrite the expression.

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

Step 3.4.2.1.4

Simplify.

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

Step 3.4.2.1.5

Apply the distributive property.

$4 y + 4 \cdot - 1 = {(x - y)}^{2}$

Step 3.4.2.1.6

Multiply $4$ by $- 1$ .

$4 y - 4 = {(x - y)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

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

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Step 3.4.3.1.1

Rewrite ${(x - y)}^{2}$ as $(x - y) (x - y)$ .

$4 y - 4 = (x - y) (x - y)$

Step 3.4.3.1.2

Expand $(x - y) (x - y)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$4 y - 4 = x (x - y) - y (x - y)$

Step 3.4.3.1.2.2

Apply the distributive property.

$4 y - 4 = x \cdot x + x (- y) - y (x - y)$

Step 3.4.3.1.2.3

Apply the distributive property.

$4 y - 4 = x \cdot x + x (- y) - y x - y (- y)$

Step 3.4.3.1.3

Simplify and combine like terms.

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Step 3.4.3.1.3.1

Simplify each term.

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Step 3.4.3.1.3.1.1

Multiply $x$ by $x$ .

$4 y - 4 = x^{2} + x (- y) - y x - y (- y)$

Step 3.4.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$4 y - 4 = x^{2} - x y - y x - y (- y)$

Step 3.4.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$4 y - 4 = x^{2} - x y - y x - 1 \cdot - 1 y \cdot y$

Step 3.4.3.1.3.1.4

Multiply $y$ by $y$ by adding the exponents.

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Step 3.4.3.1.3.1.4.1

Move $y$ .

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

Step 3.4.3.1.3.1.4.2

Multiply $y$ by $y$ .

$4 y - 4 = x^{2} - x y - y x - 1 \cdot - 1 y^{2}$

Step 3.4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.4.3.1.3.1.6

Multiply $y^{2}$ by $1$ .

$4 y - 4 = x^{2} - x y - y x + y^{2}$

Step 3.4.3.1.3.2

Subtract $y x$ from $- x y$ .

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Step 3.4.3.1.3.2.1

Move $y$ .

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

Step 3.4.3.1.3.2.2

Subtract $x y$ from $- x y$ .

$4 y - 4 = x^{2} - 2 x y + y^{2}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 2 x y + y^{2} = 4 y - 4$

Step 3.5.2

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

$x^{2} - 2 x y + y^{2} - 4 y = - 4$

Step 3.5.3

Add $4$ to both sides of the equation.

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

Step 3.5.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5.5

Substitute the values $a = 1$ , $b = - 2 x - 4$ , and $c = x^{2} + 4$ into the quadratic formula and solve for $y$ .

$\frac{- (- 2 x - 4) \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.6

Simplify.

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Step 3.5.6.1

Simplify the numerator.

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Step 3.5.6.1.1

Apply the distributive property.

$y = \frac{- (- 2 x) + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.6.1.2

Multiply $- 2$ by $- 1$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.6.1.3

Multiply $- 1$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.6.1.4

Add parentheses.

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.6.1.5

Let $u = 1 \cdot (x^{2} + 4)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 4)$ .

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Step 3.5.6.1.5.1

Rewrite ${(- 2 x - 4)}^{2}$ as $(- 2 x - 4) (- 2 x - 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{(- 2 x - 4) (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.2

Expand $(- 2 x - 4) (- 2 x - 4)$ using the FOIL Method.

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Step 3.5.6.1.5.2.1

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x - 4) - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.2.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.2.3

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3

Simplify and combine like terms.

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Step 3.5.6.1.5.3.1

Simplify each term.

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Step 3.5.6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x \cdot x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.6.1.5.3.1.2.1

Move $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x^{2}) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.4

Multiply $- 4$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.5

Multiply $- 2$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.1.6

Multiply $- 4$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x + 16 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.6.1.5.3.2

Add $8 x$ and $8 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.6.1.6

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

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Step 3.5.6.1.6.1

Factor $4$ out of $4 x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.6.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.6.1.6.3

Factor $4$ out of $16$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) - 4 u}}{2 \cdot 1}$

Step 3.5.6.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.6.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (4 x)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.6.1.6.6

Factor $4$ out of $4 (x^{2} + 4 x) + 4 (4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.6.1.6.7

Factor $4$ out of $4 (x^{2} + 4 x + 4) + 4 (- u)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - u)}}{2 \cdot 1}$

Step 3.5.6.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (1 \cdot (x^{2} + 4)))}}{2 \cdot 1}$

Step 3.5.6.1.8

Simplify.

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Step 3.5.6.1.8.1

Simplify each term.

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Step 3.5.6.1.8.1.1

Multiply $x^{2} + 4$ by $1$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.6.1.8.1.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 1 \cdot 4)}}{2 \cdot 1}$

Step 3.5.6.1.8.1.3

Multiply $- 1$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 4)}}{2 \cdot 1}$

Step 3.5.6.1.8.2

Combine the opposite terms in $x^{2} + 4 x + 4 - x^{2} - 4$ .

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Step 3.5.6.1.8.2.1

Subtract $x^{2}$ from $x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 + 0 - 4)}}{2 \cdot 1}$

Step 3.5.6.1.8.2.2

Add $4 x + 4$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 - 4)}}{2 \cdot 1}$

Step 3.5.6.1.8.2.3

Subtract $4$ from $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 0)}}{2 \cdot 1}$

Step 3.5.6.1.8.2.4

Add $4 x$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x)}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 \cdot (4 x)}}{2 \cdot 1}$

Step 3.5.6.1.9

Multiply $4$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{16 x}}{2 \cdot 1}$

Step 3.5.6.1.10

Rewrite $16 x$ as ${(2^{2})}^{2} x$ .

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Step 3.5.6.1.10.1

Rewrite $16$ as $4^{2}$ .

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

Step 3.5.6.1.10.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.5.6.1.11

Pull terms out from under the radical.

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

Step 3.5.6.1.12

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

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

Step 3.5.6.2

Multiply $2$ by $1$ .

$y = \frac{2 x + 4 \pm 4 \sqrt{x}}{2}$

Step 3.5.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.5.7.1

Simplify the numerator.

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Step 3.5.7.1.1

Apply the distributive property.

$y = \frac{- (- 2 x) + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.7.1.2

Multiply $- 2$ by $- 1$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.7.1.3

Multiply $- 1$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.7.1.4

Add parentheses.

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.7.1.5

Let $u = 1 \cdot (x^{2} + 4)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 4)$ .

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Step 3.5.7.1.5.1

Rewrite ${(- 2 x - 4)}^{2}$ as $(- 2 x - 4) (- 2 x - 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{(- 2 x - 4) (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.2

Expand $(- 2 x - 4) (- 2 x - 4)$ using the FOIL Method.

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Step 3.5.7.1.5.2.1

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x - 4) - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.2.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.2.3

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3

Simplify and combine like terms.

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Step 3.5.7.1.5.3.1

Simplify each term.

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Step 3.5.7.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x \cdot x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.7.1.5.3.1.2.1

Move $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x^{2}) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.4

Multiply $- 4$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.5

Multiply $- 2$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.1.6

Multiply $- 4$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x + 16 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.7.1.5.3.2

Add $8 x$ and $8 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.7.1.6

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

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Step 3.5.7.1.6.1

Factor $4$ out of $4 x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.7.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.7.1.6.3

Factor $4$ out of $16$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) - 4 u}}{2 \cdot 1}$

Step 3.5.7.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.7.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (4 x)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.7.1.6.6

Factor $4$ out of $4 (x^{2} + 4 x) + 4 (4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.7.1.6.7

Factor $4$ out of $4 (x^{2} + 4 x + 4) + 4 (- u)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - u)}}{2 \cdot 1}$

Step 3.5.7.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (1 \cdot (x^{2} + 4)))}}{2 \cdot 1}$

Step 3.5.7.1.8

Simplify.

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Step 3.5.7.1.8.1

Simplify each term.

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Step 3.5.7.1.8.1.1

Multiply $x^{2} + 4$ by $1$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.7.1.8.1.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 1 \cdot 4)}}{2 \cdot 1}$

Step 3.5.7.1.8.1.3

Multiply $- 1$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 4)}}{2 \cdot 1}$

Step 3.5.7.1.8.2

Combine the opposite terms in $x^{2} + 4 x + 4 - x^{2} - 4$ .

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Step 3.5.7.1.8.2.1

Subtract $x^{2}$ from $x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 + 0 - 4)}}{2 \cdot 1}$

Step 3.5.7.1.8.2.2

Add $4 x + 4$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 - 4)}}{2 \cdot 1}$

Step 3.5.7.1.8.2.3

Subtract $4$ from $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 0)}}{2 \cdot 1}$

Step 3.5.7.1.8.2.4

Add $4 x$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x)}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 \cdot (4 x)}}{2 \cdot 1}$

Step 3.5.7.1.9

Multiply $4$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{16 x}}{2 \cdot 1}$

Step 3.5.7.1.10

Rewrite $16 x$ as ${(2^{2})}^{2} x$ .

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Step 3.5.7.1.10.1

Rewrite $16$ as $4^{2}$ .

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

Step 3.5.7.1.10.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.5.7.1.11

Pull terms out from under the radical.

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

Step 3.5.7.1.12

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

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

Step 3.5.7.2

Multiply $2$ by $1$ .

$y = \frac{2 x + 4 \pm 4 \sqrt{x}}{2}$

Step 3.5.7.3

Change the $\pm$ to $+$ .

$y = \frac{2 x + 4 + 4 \sqrt{x}}{2}$

Step 3.5.7.4

Cancel the common factor of $2 x + 4 + 4 \sqrt{x}$ and $2$ .

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Step 3.5.7.4.1

Factor $2$ out of $2 x$ .

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

Step 3.5.7.4.2

Factor $2$ out of $4$ .

$y = \frac{2 (x) + 2 \cdot 2 + 4 \sqrt{x}}{2}$

Step 3.5.7.4.3

Factor $2$ out of $2 (x) + 2 (2)$ .

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

Step 3.5.7.4.4

Factor $2$ out of $4 \sqrt{x}$ .

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

Step 3.5.7.4.5

Factor $2$ out of $2 (x + 2) + 2 (2 \sqrt{x})$ .

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

Step 3.5.7.4.6

Cancel the common factors.

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Step 3.5.7.4.6.1

Factor $2$ out of $2$ .

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

Step 3.5.7.4.6.2

Cancel the common factor.

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

Step 3.5.7.4.6.3

Rewrite the expression.

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

Step 3.5.7.4.6.4

Divide $x + 2 + 2 \sqrt{x}$ by $1$ .

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

Step 3.5.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.5.8.1

Simplify the numerator.

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Step 3.5.8.1.1

Apply the distributive property.

$y = \frac{- (- 2 x) + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.8.1.2

Multiply $- 2$ by $- 1$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.8.1.3

Multiply $- 1$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot 1 \cdot (x^{2} + 4)}}{2 \cdot 1}$

Step 3.5.8.1.4

Add parentheses.

$y = \frac{2 x + 4 \pm \sqrt{{(- 2 x - 4)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.8.1.5

Let $u = 1 \cdot (x^{2} + 4)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} + 4)$ .

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Step 3.5.8.1.5.1

Rewrite ${(- 2 x - 4)}^{2}$ as $(- 2 x - 4) (- 2 x - 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{(- 2 x - 4) (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.2

Expand $(- 2 x - 4) (- 2 x - 4)$ using the FOIL Method.

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Step 3.5.8.1.5.2.1

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x - 4) - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.2.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x - 4) - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.2.3

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3

Simplify and combine like terms.

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Step 3.5.8.1.5.3.1

Simplify each term.

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Step 3.5.8.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x \cdot x) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.5.8.1.5.3.1.2.1

Move $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{2 x + 4 \pm \sqrt{- 2 \cdot (- 2 x^{2}) - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} - 2 x \cdot - 4 - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.4

Multiply $- 4$ by $- 2$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x - 4 (- 2 x) - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.5

Multiply $- 2$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x - 4 \cdot - 4 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.1.6

Multiply $- 4$ by $- 4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 8 x + 8 x + 16 - 4 \cdot u}}{2 \cdot 1}$

Step 3.5.8.1.5.3.2

Add $8 x$ and $8 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 x^{2} + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.8.1.6

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

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Step 3.5.8.1.6.1

Factor $4$ out of $4 x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 16 x + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.8.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 16 - 4 u}}{2 \cdot 1}$

Step 3.5.8.1.6.3

Factor $4$ out of $16$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) - 4 u}}{2 \cdot 1}$

Step 3.5.8.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2}) + 4 (4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.8.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (4 x)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x) + 4 (4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.8.1.6.6

Factor $4$ out of $4 (x^{2} + 4 x) + 4 (4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4) + 4 (- u)}}{2 \cdot 1}$

Step 3.5.8.1.6.7

Factor $4$ out of $4 (x^{2} + 4 x + 4) + 4 (- u)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - u)}}{2 \cdot 1}$

Step 3.5.8.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} + 4)$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (1 \cdot (x^{2} + 4)))}}{2 \cdot 1}$

Step 3.5.8.1.8

Simplify.

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Step 3.5.8.1.8.1

Simplify each term.

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Step 3.5.8.1.8.1.1

Multiply $x^{2} + 4$ by $1$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - (x^{2} + 4))}}{2 \cdot 1}$

Step 3.5.8.1.8.1.2

Apply the distributive property.

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 1 \cdot 4)}}{2 \cdot 1}$

Step 3.5.8.1.8.1.3

Multiply $- 1$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (x^{2} + 4 x + 4 - x^{2} - 4)}}{2 \cdot 1}$

Step 3.5.8.1.8.2

Combine the opposite terms in $x^{2} + 4 x + 4 - x^{2} - 4$ .

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Step 3.5.8.1.8.2.1

Subtract $x^{2}$ from $x^{2}$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 + 0 - 4)}}{2 \cdot 1}$

Step 3.5.8.1.8.2.2

Add $4 x + 4$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 4 - 4)}}{2 \cdot 1}$

Step 3.5.8.1.8.2.3

Subtract $4$ from $4$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x + 0)}}{2 \cdot 1}$

Step 3.5.8.1.8.2.4

Add $4 x$ and $0$ .

$y = \frac{2 x + 4 \pm \sqrt{4 (4 x)}}{2 \cdot 1}$

$y = \frac{2 x + 4 \pm \sqrt{4 \cdot (4 x)}}{2 \cdot 1}$

Step 3.5.8.1.9

Multiply $4$ by $4$ .

$y = \frac{2 x + 4 \pm \sqrt{16 x}}{2 \cdot 1}$

Step 3.5.8.1.10

Rewrite $16 x$ as ${(2^{2})}^{2} x$ .

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Step 3.5.8.1.10.1

Rewrite $16$ as $4^{2}$ .

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

Step 3.5.8.1.10.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.5.8.1.11

Pull terms out from under the radical.

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

Step 3.5.8.1.12

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

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

Step 3.5.8.2

Multiply $2$ by $1$ .

$y = \frac{2 x + 4 \pm 4 \sqrt{x}}{2}$

Step 3.5.8.3

Change the $\pm$ to $-$ .

$y = \frac{2 x + 4 - 4 \sqrt{x}}{2}$

Step 3.5.8.4

Cancel the common factor of $2 x + 4 - 4 \sqrt{x}$ and $2$ .

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Step 3.5.8.4.1

Factor $2$ out of $2 x$ .

$y = \frac{2 (x) + 4 - 4 \sqrt{x}}{2}$

Step 3.5.8.4.2

Factor $2$ out of $4$ .

$y = \frac{2 (x) + 2 \cdot 2 - 4 \sqrt{x}}{2}$

Step 3.5.8.4.3

Factor $2$ out of $2 (x) + 2 (2)$ .

$y = \frac{2 (x + 2) - 4 \sqrt{x}}{2}$

Step 3.5.8.4.4

Factor $2$ out of $- 4 \sqrt{x}$ .

$y = \frac{2 (x + 2) + 2 (- 2 \sqrt{x})}{2}$

Step 3.5.8.4.5

Factor $2$ out of $2 (x + 2) + 2 (- 2 \sqrt{x})$ .

$y = \frac{2 (x + 2 - 2 \sqrt{x})}{2}$

Step 3.5.8.4.6

Cancel the common factors.

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Step 3.5.8.4.6.1

Factor $2$ out of $2$ .

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

Step 3.5.8.4.6.2

Cancel the common factor.

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

Step 3.5.8.4.6.3

Rewrite the expression.

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

Step 3.5.8.4.6.4

Divide $x + 2 - 2 \sqrt{x}$ by $1$ .

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

Step 3.5.9

The final answer is the combination of both solutions.

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

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

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

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

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

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

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 5

Verify if $f^{- 1} (x) = x + 2 + 2 \sqrt{x}, x + 2 - 2 \sqrt{x}$ is the inverse of $f (x) = x + 2 \sqrt{x - 1}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x + 2 \sqrt{x - 1}$ and $f^{- 1} (x) = x + 2 + 2 \sqrt{x}, x + 2 - 2 \sqrt{x}$ and compare them.

Step 5.2

Find the range of $f (x) = x + 2 \sqrt{x - 1}$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[1, \infty)$

Step 5.3

Find the domain of $x + 2 + 2 \sqrt{x}$ .

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Step 5.3.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 5.3.2

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = x + 2 + 2 \sqrt{x}, x + 2 - 2 \sqrt{x}$ is not equal to the range of $f (x) = x + 2 \sqrt{x - 1}$ , then $f^{- 1} (x) = x + 2 + 2 \sqrt{x}, x + 2 - 2 \sqrt{x}$ is not an inverse of $f (x) = x + 2 \sqrt{x - 1}$ .

There is no inverse

Step 6