Algebra Examples

$x (y + 2) = {(y + 2)}^{2} + 1$

Step 1

Simplify $x (y + 2)$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

$x (y + 2) = {(y + 2)}^{2} + 1$

Step 1.3

Apply the distributive property.

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

Step 1.4

Move $2$ to the left of $x$ .

$x y + 2 x = {(y + 2)}^{2} + 1$

Step 2

Simplify ${(y + 2)}^{2} + 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$x y + 2 x = (y + 2) (y + 2) + 1$

Step 2.1.2

Expand $(y + 2) (y + 2)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x y + 2 x = y (y + 2) + 2 (y + 2) + 1$

Step 2.1.2.2

Apply the distributive property.

$x y + 2 x = y \cdot y + y \cdot 2 + 2 (y + 2) + 1$

Step 2.1.2.3

Apply the distributive property.

$x y + 2 x = y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2 + 1$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $y$ by $y$ .

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

Step 2.1.3.1.2

Move $2$ to the left of $y$ .

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

Step 2.1.3.1.3

Multiply $2$ by $2$ .

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

Step 2.1.3.2

Add $2 y$ and $2 y$ .

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

Step 2.2

Add $4$ and $1$ .

$x y + 2 x = y^{2} + 4 y + 5$

Step 3

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

$y^{2} + 4 y + 5 = x y + 2 x$

Step 4

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

$y^{2} + 4 y + 5 - x y = 2 x$

Step 5

Subtract $2 x$ from both sides of the equation.

$y^{2} + 4 y + 5 - x y - 2 x = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = 4 - x$ , and $c = 5 - 2 x$ into the quadratic formula and solve for $y$ .

$\frac{- (4 - x) \pm \sqrt{{(4 - x)}^{2} - 4 \cdot (1 \cdot (5 - 2 x))}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.3

Multiply $- - x$ .

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Step 8.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + 1 x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.3.2

Multiply $x$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.4

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

$y = \frac{- 4 + x \pm \sqrt{(4 - x) (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.5

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

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Step 8.1.5.1

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 (4 - x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.5.2

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.5.3

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6

Simplify and combine like terms.

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Step 8.1.6.1

Simplify each term.

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Step 8.1.6.1.1

Multiply $4$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x \cdot x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.5

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

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Step 8.1.6.1.5.1

Move $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 (x \cdot x)) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.5.2

Multiply $x$ by $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x^{2}) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + 1 x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.1.7

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

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.6.2

Subtract $4 x$ from $- 4 x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 8.1.8

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 5 - 4 (- 2 x)}}{2 \cdot 1}$

Step 8.1.9

Multiply $- 4$ by $5$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 - 4 (- 2 x)}}{2 \cdot 1}$

Step 8.1.10

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 + 8 x}}{2 \cdot 1}$

Step 8.1.11

Subtract $20$ from $16$ .

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

Step 8.1.12

Add $- 8 x$ and $8 x$ .

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

Step 8.1.13

Add $x^{2} - 4$ and $0$ .

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

Step 8.1.14

Rewrite $x^{2} - 4$ in a factored form.

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Step 8.1.14.1

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

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

Step 8.1.14.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 8.2

Multiply $2$ by $1$ .

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

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.3

Multiply $- - x$ .

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Step 9.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + 1 x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.3.2

Multiply $x$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.4

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

$y = \frac{- 4 + x \pm \sqrt{(4 - x) (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.5

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

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Step 9.1.5.1

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 (4 - x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.5.2

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.5.3

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6

Simplify and combine like terms.

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Step 9.1.6.1

Simplify each term.

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Step 9.1.6.1.1

Multiply $4$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x \cdot x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.5

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

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Step 9.1.6.1.5.1

Move $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 (x \cdot x)) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.5.2

Multiply $x$ by $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x^{2}) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + 1 x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.1.7

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

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.6.2

Subtract $4 x$ from $- 4 x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 9.1.8

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 5 - 4 (- 2 x)}}{2 \cdot 1}$

Step 9.1.9

Multiply $- 4$ by $5$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 - 4 (- 2 x)}}{2 \cdot 1}$

Step 9.1.10

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 + 8 x}}{2 \cdot 1}$

Step 9.1.11

Subtract $20$ from $16$ .

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

Step 9.1.12

Add $- 8 x$ and $8 x$ .

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

Step 9.1.13

Add $x^{2} - 4$ and $0$ .

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

Step 9.1.14

Rewrite $x^{2} - 4$ in a factored form.

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Step 9.1.14.1

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

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

Step 9.1.14.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 9.2

Multiply $2$ by $1$ .

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

Step 9.3

Change the $\pm$ to $+$ .

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

Step 9.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 9.5

Factor $- 1$ out of $x$ .

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

Step 9.6

Factor $- 1$ out of $- 1 (4) - 1 (- x)$ .

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

Step 9.7

Factor $- 1$ out of $\sqrt{(x + 2) (x - 2)}$ .

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

Step 9.8

Factor $- 1$ out of $- 1 (4 - x) - 1 (- \sqrt{(x + 2) (x - 2)})$ .

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

Step 9.9

Move the negative in front of the fraction.

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

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Apply the distributive property.

$y = \frac{- 1 \cdot 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.3

Multiply $- - x$ .

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Step 10.1.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + 1 x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.3.2

Multiply $x$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{{(4 - x)}^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.4

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

$y = \frac{- 4 + x \pm \sqrt{(4 - x) (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.5

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

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Step 10.1.5.1

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 (4 - x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.5.2

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x (4 - x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.5.3

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6

Simplify and combine like terms.

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Step 10.1.6.1

Simplify each term.

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Step 10.1.6.1.1

Multiply $4$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 + 4 (- x) - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.2

Multiply $- 1$ by $4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - x \cdot 4 - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.3

Multiply $4$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - x (- x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.4

Rewrite using the commutative property of multiplication.

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x \cdot x) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.5

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

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Step 10.1.6.1.5.1

Move $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 (x \cdot x)) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.5.2

Multiply $x$ by $x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x - 1 \cdot (- 1 x^{2}) - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + 1 x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.1.7

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

$y = \frac{- 4 + x \pm \sqrt{16 - 4 x - 4 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.6.2

Subtract $4 x$ from $- 4 x$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 1 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.7

Multiply $- 4$ by $1$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot (5 - 2 x)}}{2 \cdot 1}$

Step 10.1.8

Apply the distributive property.

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 4 \cdot 5 - 4 (- 2 x)}}{2 \cdot 1}$

Step 10.1.9

Multiply $- 4$ by $5$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 - 4 (- 2 x)}}{2 \cdot 1}$

Step 10.1.10

Multiply $- 2$ by $- 4$ .

$y = \frac{- 4 + x \pm \sqrt{16 - 8 x + x^{2} - 20 + 8 x}}{2 \cdot 1}$

Step 10.1.11

Subtract $20$ from $16$ .

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

Step 10.1.12

Add $- 8 x$ and $8 x$ .

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

Step 10.1.13

Add $x^{2} - 4$ and $0$ .

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

Step 10.1.14

Rewrite $x^{2} - 4$ in a factored form.

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Step 10.1.14.1

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

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

Step 10.1.14.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 10.2

Multiply $2$ by $1$ .

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

Step 10.3

Change the $\pm$ to $-$ .

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

Step 10.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 10.5

Factor $- 1$ out of $x$ .

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

Step 10.6

Factor $- 1$ out of $- 1 (4) - 1 (- x)$ .

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

Step 10.7

Factor $- 1$ out of $- \sqrt{(x + 2) (x - 2)}$ .

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

Step 10.8

Factor $- 1$ out of $- 1 (4 - x) - (\sqrt{(x + 2) (x - 2)})$ .

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

Step 10.9

Move the negative in front of the fraction.

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

Step 11

The final answer is the combination of both solutions.

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

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