Algebra Examples

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

Step 1

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

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

Step 2

Subtract $x$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

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

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

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

Step 4.2.1.3.2.2

Rewrite the expression.

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

Step 4.2.1.4

Simplify.

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

Step 4.2.1.5

Apply the distributive property.

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

Step 4.2.1.6

Multiply $4$ by $- 1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

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

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Step 4.3.1.2.1

Apply the distributive property.

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

Step 4.3.1.2.2

Apply the distributive property.

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

Step 4.3.1.2.3

Apply the distributive property.

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

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $y$ by $y$ .

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

Step 4.3.1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.3.1.4

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

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Step 4.3.1.3.1.4.1

Move $x$ .

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

Step 4.3.1.3.1.4.2

Multiply $x$ by $x$ .

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

Step 4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.3.1.6

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

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

Step 4.3.1.3.2

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

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Step 4.3.1.3.2.1

Move $y$ .

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

Step 4.3.1.3.2.2

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

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

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Add $4$ to both sides of the equation.

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

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

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

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

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Apply the distributive property.

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

Step 5.6.1.2

Multiply $- 2$ by $- 1$ .

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

Step 5.6.1.3

Multiply $- 1$ by $- 4$ .

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

Step 5.6.1.4

Add parentheses.

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

Step 5.6.1.5

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

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Step 5.6.1.5.1

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

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

Step 5.6.1.5.2

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

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Step 5.6.1.5.2.1

Apply the distributive property.

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

Step 5.6.1.5.2.2

Apply the distributive property.

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

Step 5.6.1.5.2.3

Apply the distributive property.

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

Step 5.6.1.5.3

Simplify and combine like terms.

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Step 5.6.1.5.3.1

Simplify each term.

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Step 5.6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.6.1.5.3.1.2

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

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Step 5.6.1.5.3.1.2.1

Move $y$ .

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

Step 5.6.1.5.3.1.2.2

Multiply $y$ by $y$ .

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

Step 5.6.1.5.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 5.6.1.5.3.1.4

Multiply $- 4$ by $- 2$ .

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

Step 5.6.1.5.3.1.5

Multiply $- 2$ by $- 4$ .

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

Step 5.6.1.5.3.1.6

Multiply $- 4$ by $- 4$ .

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

Step 5.6.1.5.3.2

Add $8 y$ and $8 y$ .

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

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

Step 5.6.1.6

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

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Step 5.6.1.6.1

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

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

Step 5.6.1.6.2

Factor $4$ out of $16 y$ .

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

Step 5.6.1.6.3

Factor $4$ out of $16$ .

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

Step 5.6.1.6.4

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

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

Step 5.6.1.6.5

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

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

Step 5.6.1.6.6

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

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

Step 5.6.1.6.7

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

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

Step 5.6.1.7

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

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

Step 5.6.1.8

Simplify.

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Step 5.6.1.8.1

Simplify each term.

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Step 5.6.1.8.1.1

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

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

Step 5.6.1.8.1.2

Apply the distributive property.

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

Step 5.6.1.8.1.3

Multiply $- 1$ by $4$ .

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

Step 5.6.1.8.2

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

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Step 5.6.1.8.2.1

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

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

Step 5.6.1.8.2.2

Add $4 y + 4$ and $0$ .

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

Step 5.6.1.8.2.3

Subtract $4$ from $4$ .

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

Step 5.6.1.8.2.4

Add $4 y$ and $0$ .

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

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

Step 5.6.1.9

Multiply $4$ by $4$ .

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

Step 5.6.1.10

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

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Step 5.6.1.10.1

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

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

Step 5.6.1.10.2

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

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

Step 5.6.1.11

Pull terms out from under the radical.

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

Step 5.6.1.12

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

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

Step 5.6.2

Multiply $2$ by $1$ .

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

Step 5.7

The final answer is the combination of both solutions.

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

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

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

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