Precalculus Examples

$x^{2} + 2 x y + y^{2} - 8 x + 8 y = 0$

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 1$ , $b = 2 x + 8$ , and $c = x^{2} - 8 x$ into the quadratic formula and solve for $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Apply the distributive property.

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

Step 1.3.1.2

Multiply $2$ by $- 1$ .

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

Step 1.3.1.3

Multiply $- 1$ by $8$ .

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

Step 1.3.1.4

Add parentheses.

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

Step 1.3.1.5

Let $u = 1 \cdot (x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8 x)$ .

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Step 1.3.1.5.1

Rewrite ${(2 x + 8)}^{2}$ as $(2 x + 8) (2 x + 8)$ .

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

Step 1.3.1.5.2

Expand $(2 x + 8) (2 x + 8)$ using the FOIL Method.

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Step 1.3.1.5.2.1

Apply the distributive property.

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

Step 1.3.1.5.2.2

Apply the distributive property.

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

Step 1.3.1.5.2.3

Apply the distributive property.

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

Step 1.3.1.5.3

Simplify and combine like terms.

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Step 1.3.1.5.3.1

Simplify each term.

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Step 1.3.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.5.3.1.2

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

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Step 1.3.1.5.3.1.2.1

Move $x$ .

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

Step 1.3.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.5.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.5.3.1.4

Multiply $8$ by $2$ .

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

Step 1.3.1.5.3.1.5

Multiply $2$ by $8$ .

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

Step 1.3.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 16 x + 16 x + 64 - 4 \cdot u}}{2 \cdot 1}$

Step 1.3.1.5.3.2

Add $16 x$ and $16 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.3.1.6

Factor $4$ out of $4 x^{2} + 32 x + 64 - 4 u$ .

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Step 1.3.1.6.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.3.1.6.2

Factor $4$ out of $32 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (8 x) + 64 - 4 u}}{2 \cdot 1}$

Step 1.3.1.6.3

Factor $4$ out of $64$ .

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

Step 1.3.1.6.4

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

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

Step 1.3.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (8 x)$ .

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

Step 1.3.1.6.6

Factor $4$ out of $4 (x^{2} + 8 x) + 4 (16)$ .

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

Step 1.3.1.6.7

Factor $4$ out of $4 (x^{2} + 8 x + 16) + 4 (- u)$ .

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

Step 1.3.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8 x)$ .

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

Step 1.3.1.8

Simplify.

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Step 1.3.1.8.1

Simplify each term.

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Step 1.3.1.8.1.1

Multiply $x^{2} - 8 x$ by $1$ .

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

Step 1.3.1.8.1.2

Apply the distributive property.

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

Step 1.3.1.8.1.3

Multiply $- 8$ by $- 1$ .

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

Step 1.3.1.8.2

Combine the opposite terms in $x^{2} + 8 x + 16 - x^{2} + 8 x$ .

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Step 1.3.1.8.2.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (8 x + 16 + 0 + 8 x)}}{2 \cdot 1}$

Step 1.3.1.8.2.2

Add $8 x + 16$ and $0$ .

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

Step 1.3.1.8.3

Add $8 x$ and $8 x$ .

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

Step 1.3.1.9

Factor $16$ out of $16 x + 16$ .

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Step 1.3.1.9.1

Factor $16$ out of $16 x$ .

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

Step 1.3.1.9.2

Factor $16$ out of $16$ .

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

Step 1.3.1.9.3

Factor $16$ out of $16 (x) + 16 (1)$ .

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

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

Step 1.3.1.10

Multiply $4$ by $16$ .

$y = \frac{- 2 x - 8 \pm \sqrt{64 (x + 1)}}{2 \cdot 1}$

Step 1.3.1.11

Rewrite $64 (x + 1)$ as $8^{2} (x + 1^{2})$ .

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Step 1.3.1.11.1

Rewrite $64$ as $8^{2}$ .

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

Step 1.3.1.11.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.3.1.12

Pull terms out from under the radical.

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

Step 1.3.1.13

One to any power is one.

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

Step 1.3.2

Multiply $2$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Apply the distributive property.

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

Step 1.4.1.2

Multiply $2$ by $- 1$ .

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

Step 1.4.1.3

Multiply $- 1$ by $8$ .

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

Step 1.4.1.4

Add parentheses.

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

Step 1.4.1.5

Let $u = 1 \cdot (x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8 x)$ .

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Step 1.4.1.5.1

Rewrite ${(2 x + 8)}^{2}$ as $(2 x + 8) (2 x + 8)$ .

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

Step 1.4.1.5.2

Expand $(2 x + 8) (2 x + 8)$ using the FOIL Method.

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Step 1.4.1.5.2.1

Apply the distributive property.

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

Step 1.4.1.5.2.2

Apply the distributive property.

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

Step 1.4.1.5.2.3

Apply the distributive property.

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

Step 1.4.1.5.3

Simplify and combine like terms.

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Step 1.4.1.5.3.1

Simplify each term.

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Step 1.4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.1.5.3.1.2

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

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Step 1.4.1.5.3.1.2.1

Move $x$ .

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

Step 1.4.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.4.1.5.3.1.3

Multiply $2$ by $2$ .

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

Step 1.4.1.5.3.1.4

Multiply $8$ by $2$ .

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

Step 1.4.1.5.3.1.5

Multiply $2$ by $8$ .

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

Step 1.4.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 16 x + 16 x + 64 - 4 \cdot u}}{2 \cdot 1}$

Step 1.4.1.5.3.2

Add $16 x$ and $16 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.4.1.6

Factor $4$ out of $4 x^{2} + 32 x + 64 - 4 u$ .

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Step 1.4.1.6.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.4.1.6.2

Factor $4$ out of $32 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (8 x) + 64 - 4 u}}{2 \cdot 1}$

Step 1.4.1.6.3

Factor $4$ out of $64$ .

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

Step 1.4.1.6.4

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

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

Step 1.4.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (8 x)$ .

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

Step 1.4.1.6.6

Factor $4$ out of $4 (x^{2} + 8 x) + 4 (16)$ .

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

Step 1.4.1.6.7

Factor $4$ out of $4 (x^{2} + 8 x + 16) + 4 (- u)$ .

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

Step 1.4.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8 x)$ .

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

Step 1.4.1.8

Simplify.

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Step 1.4.1.8.1

Simplify each term.

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Step 1.4.1.8.1.1

Multiply $x^{2} - 8 x$ by $1$ .

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

Step 1.4.1.8.1.2

Apply the distributive property.

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

Step 1.4.1.8.1.3

Multiply $- 8$ by $- 1$ .

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

Step 1.4.1.8.2

Combine the opposite terms in $x^{2} + 8 x + 16 - x^{2} + 8 x$ .

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Step 1.4.1.8.2.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (8 x + 16 + 0 + 8 x)}}{2 \cdot 1}$

Step 1.4.1.8.2.2

Add $8 x + 16$ and $0$ .

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

Step 1.4.1.8.3

Add $8 x$ and $8 x$ .

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

Step 1.4.1.9

Factor $16$ out of $16 x + 16$ .

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Step 1.4.1.9.1

Factor $16$ out of $16 x$ .

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

Step 1.4.1.9.2

Factor $16$ out of $16$ .

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

Step 1.4.1.9.3

Factor $16$ out of $16 (x) + 16 (1)$ .

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

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

Step 1.4.1.10

Multiply $4$ by $16$ .

$y = \frac{- 2 x - 8 \pm \sqrt{64 (x + 1)}}{2 \cdot 1}$

Step 1.4.1.11

Rewrite $64 (x + 1)$ as $8^{2} (x + 1^{2})$ .

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Step 1.4.1.11.1

Rewrite $64$ as $8^{2}$ .

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

Step 1.4.1.11.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.4.1.12

Pull terms out from under the radical.

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

Step 1.4.1.13

One to any power is one.

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

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

Change the $\pm$ to $+$ .

$y = \frac{- 2 x - 8 + 8 \sqrt{x + 1}}{2}$

Step 1.4.4

Cancel the common factor of $- 2 x - 8 + 8 \sqrt{x + 1}$ and $2$ .

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Step 1.4.4.1

Factor $2$ out of $- 2 x$ .

$y = \frac{2 (- x) - 8 + 8 \sqrt{x + 1}}{2}$

Step 1.4.4.2

Factor $2$ out of $- 8$ .

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

Step 1.4.4.3

Factor $2$ out of $2 (- x) + 2 (- 4)$ .

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

Step 1.4.4.4

Factor $2$ out of $8 \sqrt{x + 1}$ .

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

Step 1.4.4.5

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

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

Step 1.4.4.6

Cancel the common factors.

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Step 1.4.4.6.1

Factor $2$ out of $2$ .

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

Step 1.4.4.6.2

Cancel the common factor.

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

Step 1.4.4.6.3

Rewrite the expression.

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

Step 1.4.4.6.4

Divide $- x - 4 + 4 \sqrt{x + 1}$ by $1$ .

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

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Apply the distributive property.

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

Step 1.5.1.2

Multiply $2$ by $- 1$ .

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

Step 1.5.1.3

Multiply $- 1$ by $8$ .

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

Step 1.5.1.4

Add parentheses.

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

Step 1.5.1.5

Let $u = 1 \cdot (x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $1 \cdot (x^{2} - 8 x)$ .

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Step 1.5.1.5.1

Rewrite ${(2 x + 8)}^{2}$ as $(2 x + 8) (2 x + 8)$ .

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

Step 1.5.1.5.2

Expand $(2 x + 8) (2 x + 8)$ using the FOIL Method.

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Step 1.5.1.5.2.1

Apply the distributive property.

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

Step 1.5.1.5.2.2

Apply the distributive property.

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

Step 1.5.1.5.2.3

Apply the distributive property.

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

Step 1.5.1.5.3

Simplify and combine like terms.

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Step 1.5.1.5.3.1

Simplify each term.

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Step 1.5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.1.5.3.1.2

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

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Step 1.5.1.5.3.1.2.1

Move $x$ .

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

Step 1.5.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.5.1.5.3.1.3

Multiply $2$ by $2$ .

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

Step 1.5.1.5.3.1.4

Multiply $8$ by $2$ .

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

Step 1.5.1.5.3.1.5

Multiply $2$ by $8$ .

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

Step 1.5.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 16 x + 16 x + 64 - 4 \cdot u}}{2 \cdot 1}$

Step 1.5.1.5.3.2

Add $16 x$ and $16 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 \cdot u}}{2 \cdot 1}$

$y = \frac{- 2 x - 8 \pm \sqrt{4 x^{2} + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.5.1.6

Factor $4$ out of $4 x^{2} + 32 x + 64 - 4 u$ .

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Step 1.5.1.6.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 32 x + 64 - 4 u}}{2 \cdot 1}$

Step 1.5.1.6.2

Factor $4$ out of $32 x$ .

$y = \frac{- 2 x - 8 \pm \sqrt{4 (x^{2}) + 4 (8 x) + 64 - 4 u}}{2 \cdot 1}$

Step 1.5.1.6.3

Factor $4$ out of $64$ .

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

Step 1.5.1.6.4

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

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

Step 1.5.1.6.5

Factor $4$ out of $4 (x^{2}) + 4 (8 x)$ .

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

Step 1.5.1.6.6

Factor $4$ out of $4 (x^{2} + 8 x) + 4 (16)$ .

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

Step 1.5.1.6.7

Factor $4$ out of $4 (x^{2} + 8 x + 16) + 4 (- u)$ .

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

Step 1.5.1.7

Replace all occurrences of $u$ with $1 \cdot (x^{2} - 8 x)$ .

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

Step 1.5.1.8

Simplify.

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Step 1.5.1.8.1

Simplify each term.

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Step 1.5.1.8.1.1

Multiply $x^{2} - 8 x$ by $1$ .

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

Step 1.5.1.8.1.2

Apply the distributive property.

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

Step 1.5.1.8.1.3

Multiply $- 8$ by $- 1$ .

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

Step 1.5.1.8.2

Combine the opposite terms in $x^{2} + 8 x + 16 - x^{2} + 8 x$ .

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Step 1.5.1.8.2.1

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

$y = \frac{- 2 x - 8 \pm \sqrt{4 (8 x + 16 + 0 + 8 x)}}{2 \cdot 1}$

Step 1.5.1.8.2.2

Add $8 x + 16$ and $0$ .

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

Step 1.5.1.8.3

Add $8 x$ and $8 x$ .

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

Step 1.5.1.9

Factor $16$ out of $16 x + 16$ .

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Step 1.5.1.9.1

Factor $16$ out of $16 x$ .

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

Step 1.5.1.9.2

Factor $16$ out of $16$ .

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

Step 1.5.1.9.3

Factor $16$ out of $16 (x) + 16 (1)$ .

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

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

Step 1.5.1.10

Multiply $4$ by $16$ .

$y = \frac{- 2 x - 8 \pm \sqrt{64 (x + 1)}}{2 \cdot 1}$

Step 1.5.1.11

Rewrite $64 (x + 1)$ as $8^{2} (x + 1^{2})$ .

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Step 1.5.1.11.1

Rewrite $64$ as $8^{2}$ .

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

Step 1.5.1.11.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.5.1.12

Pull terms out from under the radical.

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

Step 1.5.1.13

One to any power is one.

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

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.5.3

Change the $\pm$ to $-$ .

$y = \frac{- 2 x - 8 - 8 \sqrt{x + 1}}{2}$

Step 1.5.4

Cancel the common factor of $- 2 x - 8 - 8 \sqrt{x + 1}$ and $2$ .

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Step 1.5.4.1

Factor $2$ out of $- 2 x$ .

$y = \frac{2 (- x) - 8 - 8 \sqrt{x + 1}}{2}$

Step 1.5.4.2

Factor $2$ out of $- 8$ .

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

Step 1.5.4.3

Factor $2$ out of $2 (- x) + 2 (- 4)$ .

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

Step 1.5.4.4

Factor $2$ out of $- 8 \sqrt{x + 1}$ .

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

Step 1.5.4.5

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

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

Step 1.5.4.6

Cancel the common factors.

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Step 1.5.4.6.1

Factor $2$ out of $2$ .

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

Step 1.5.4.6.2

Cancel the common factor.

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

Step 1.5.4.6.3

Rewrite the expression.

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

Step 1.5.4.6.4

Divide $- x - 4 - 4 \sqrt{x + 1}$ by $1$ .

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

Step 1.6

The final answer is the combination of both solutions.

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

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

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

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

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

The standard form is $- x - 4 + 4 \sqrt{x + 1}, - x - 4 - 4 \sqrt{x + 1}$ .

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

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

Step 4