Linear Algebra Examples

z = 4 x - 4 y - x^{2} - y^{2}

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

Step 1

Rewrite the equation as

4 x - 4 y - x^{2} - y^{2} = z

$4 x - 4 y - x^{2} - y^{2} = z$ .

4 x - 4 y - x^{2} - y^{2} = z

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

Step 2

Subtract

z

$z$ from both sides of the equation.

4 x - 4 y - x^{2} - y^{2} - z = 0

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

Step 3

Use the quadratic formula to find the solutions.

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

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

Step 4

Substitute the values

a = - 1

$a = - 1$ ,

b = - 4

$b = - 4$ , and

c = 4 x - x^{2} - z

$c = 4 x - x^{2} - z$ into the quadratic formula and solve for

y

$y$ .

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)

${(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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Step 5.1.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2}

${(- 4)}^{2}$ .

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

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

Step 5.1.1.2

Factor

- 4

$- 4$ out of

- 4 \cdot - 1 \cdot (4 x - x^{2} - z)

$- 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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

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

Step 5.1.1.3

Factor

- 4

$- 4$ out of

- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))

$- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))$ .

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

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

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

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

Step 5.1.2

Factor

- 1

$- 1$ out of

- 4 - 1 \cdot (4 x - x^{2} - z)

$- 4 - 1 \cdot (4 x - x^{2} - z)$ .

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Step 5.1.2.1

Reorder the expression.

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Step 5.1.2.1.1

Move

4 x

$4 x$ .

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

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

Step 5.1.2.1.2

Reorder

- 4

$- 4$ and

- 1 \cdot (- x^{2} - z + 4 x)

$- 1 \cdot (- x^{2} - z + 4 x)$ .

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

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

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

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

Step 5.1.2.2

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 5.1.2.3

Factor

- 1

$- 1$ out of

- 1 (- x^{2} - z + 4 x) - 1 (4)

$- 1 (- x^{2} - z + 4 x) - 1 (4)$ .

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

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

Step 5.1.2.4

Rewrite

- 1 (- x^{2} - z + 4 x + 4)

$- 1 (- x^{2} - z + 4 x + 4)$ as

- (- x^{2} - z + 4 x + 4)

$- (- x^{2} - z + 4 x + 4)$ .

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

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

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

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

Step 5.1.3

Combine exponents.

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Step 5.1.3.1

Factor out negative.

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

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

Step 5.1.3.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

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

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

Step 5.1.4

Rewrite

4 (- x^{2} - z + 4 x + 4)

$4 (- x^{2} - z + 4 x + 4)$ as

2^{2} (- x^{2} - z + 2^{2} x + 4)

$2^{2} (- x^{2} - z + 2^{2} x + 4)$ .

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Step 5.1.4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 5.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 5.1.5

Pull terms out from under the radical.

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

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

Step 5.1.6

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 5.2

Multiply

2

$2$ by

- 1

$- 1$ .

y = \frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}

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

Step 5.3

Simplify

\frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}

$\frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}$ .

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

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

Step 5.4

Move the negative one from the denominator of

\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}

$\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}$ .

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

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

Step 5.5

Rewrite

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

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

- (2 \pm \sqrt{- x^{2} - z + 4 x + 4})

$- (2 \pm \sqrt{- x^{2} - z + 4 x + 4})$ .

y = - (2 \pm \sqrt{- x^{2} - z + 4 x + 4})

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

y = - (2 \pm \sqrt{- x^{2} - z + 4 x + 4})

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

Step 6

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)

${(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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Step 6.1.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2}

${(- 4)}^{2}$ .

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

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

Step 6.1.1.2

Factor

- 4

$- 4$ out of

- 4 \cdot - 1 \cdot (4 x - x^{2} - z)

$- 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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

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

Step 6.1.1.3

Factor

- 4

$- 4$ out of

- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))

$- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))$ .

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

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

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

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

Step 6.1.2

Factor

- 1

$- 1$ out of

- 4 - 1 \cdot (4 x - x^{2} - z)

$- 4 - 1 \cdot (4 x - x^{2} - z)$ .

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Step 6.1.2.1

Reorder the expression.

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Step 6.1.2.1.1

Move

4 x

$4 x$ .

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

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

Step 6.1.2.1.2

Reorder

- 4

$- 4$ and

- 1 \cdot (- x^{2} - z + 4 x)

$- 1 \cdot (- x^{2} - z + 4 x)$ .

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

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

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

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

Step 6.1.2.2

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 6.1.2.3

Factor

- 1

$- 1$ out of

- 1 (- x^{2} - z + 4 x) - 1 (4)

$- 1 (- x^{2} - z + 4 x) - 1 (4)$ .

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

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

Step 6.1.2.4

Rewrite

- 1 (- x^{2} - z + 4 x + 4)

$- 1 (- x^{2} - z + 4 x + 4)$ as

- (- x^{2} - z + 4 x + 4)

$- (- x^{2} - z + 4 x + 4)$ .

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

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

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

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

Step 6.1.3

Combine exponents.

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Step 6.1.3.1

Factor out negative.

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

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

Step 6.1.3.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

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

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

Step 6.1.4

Rewrite

4 (- x^{2} - z + 4 x + 4)

$4 (- x^{2} - z + 4 x + 4)$ as

2^{2} (- x^{2} - z + 2^{2} x + 4)

$2^{2} (- x^{2} - z + 2^{2} x + 4)$ .

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Step 6.1.4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 6.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 6.1.5

Pull terms out from under the radical.

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

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

Step 6.1.6

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 6.2

Multiply

2

$2$ by

- 1

$- 1$ .

y = \frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}

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

Step 6.3

Simplify

\frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}

$\frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}$ .

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

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

Step 6.4

Move the negative one from the denominator of

\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}

$\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}$ .

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

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

Step 6.5

Rewrite

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

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

- (2 \pm \sqrt{- x^{2} - z + 4 x + 4})

$- (2 \pm \sqrt{- x^{2} - z + 4 x + 4})$ .

y = - (2 \pm \sqrt{- x^{2} - z + 4 x + 4})

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

Step 6.6

Change the

\pm

$\pm$ to

+

$+$ .

y = - (2 + \sqrt{- x^{2} - z + 4 x + 4})

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

Step 6.7

Apply the distributive property.

y = - 1 \cdot 2 - \sqrt{- x^{2} - z + 4 x + 4}

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

Step 6.8

Multiply

- 1

$- 1$ by

2

$2$ .

y = - 2 - \sqrt{- x^{2} - z + 4 x + 4}

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

y = - 2 - \sqrt{- x^{2} - z + 4 x + 4}

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

Step 7

Simplify the expression to solve for the

-

$-$ portion of the

\pm

$\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)

${(- 4)}^{2} - 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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Step 7.1.1.1

Factor

- 4

$- 4$ out of

{(- 4)}^{2}

${(- 4)}^{2}$ .

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

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

Step 7.1.1.2

Factor

- 4

$- 4$ out of

- 4 \cdot - 1 \cdot (4 x - x^{2} - z)

$- 4 \cdot - 1 \cdot (4 x - x^{2} - z)$ .

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

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

Step 7.1.1.3

Factor

- 4

$- 4$ out of

- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))

$- 4 \cdot - 4 - 4 (- 1 \cdot (4 x - x^{2} - z))$ .

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

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

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

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

Step 7.1.2

Factor

- 1

$- 1$ out of

- 4 - 1 \cdot (4 x - x^{2} - z)

$- 4 - 1 \cdot (4 x - x^{2} - z)$ .

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Step 7.1.2.1

Reorder the expression.

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Step 7.1.2.1.1

Move

4 x

$4 x$ .

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

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

Step 7.1.2.1.2

Reorder

- 4

$- 4$ and

- 1 \cdot (- x^{2} - z + 4 x)

$- 1 \cdot (- x^{2} - z + 4 x)$ .

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

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

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

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

Step 7.1.2.2

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 7.1.2.3

Factor

- 1

$- 1$ out of

- 1 (- x^{2} - z + 4 x) - 1 (4)

$- 1 (- x^{2} - z + 4 x) - 1 (4)$ .

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

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

Step 7.1.2.4

Rewrite

- 1 (- x^{2} - z + 4 x + 4)

$- 1 (- x^{2} - z + 4 x + 4)$ as

- (- x^{2} - z + 4 x + 4)

$- (- x^{2} - z + 4 x + 4)$ .

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

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

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

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

Step 7.1.3

Combine exponents.

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Step 7.1.3.1

Factor out negative.

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

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

Step 7.1.3.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

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

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

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

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

Step 7.1.4

Rewrite

4 (- x^{2} - z + 4 x + 4)

$4 (- x^{2} - z + 4 x + 4)$ as

2^{2} (- x^{2} - z + 2^{2} x + 4)

$2^{2} (- x^{2} - z + 2^{2} x + 4)$ .

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Step 7.1.4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 7.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

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

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

Step 7.1.5

Pull terms out from under the radical.

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

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

Step 7.1.6

Raise

$2$ to the power of

$2$ .

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

Step 7.2

Multiply

$2$ by

$- 1$ .

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

Step 7.3

Simplify

$\frac{4 \pm 2 \sqrt{- x^{2} - z + 4 x + 4}}{- 2}$ .

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

Step 7.4

Move the negative one from the denominator of

$\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}$ .

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

Step 7.5

Rewrite

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

$- (2 \pm \sqrt{- x^{2} - z + 4 x + 4})$ .

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

Step 7.6

Change the

$\pm$ to

$-$ .

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

Step 7.7

Apply the distributive property.

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

Step 7.8

Multiply

$- 1$ by

$2$ .

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

Step 7.9

Multiply

$- - \sqrt{- x^{2} - z + 4 x + 4}$ .

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Step 7.9.1

Multiply

$- 1$ by

$- 1$ .

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

Step 7.9.2

Multiply

$\sqrt{- x^{2} - z + 4 x + 4}$ by

$1$ .

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

Step 8

The final answer is the combination of both solutions.

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

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

Step 9

Set the radicand in

$\sqrt{- x^{2} - z + 4 x + 4}$ greater than or equal to

$0$ to find where the expression is defined.

$- x^{2} - z + 4 x + 4 \geq 0$

Step 10

Solve for

$x$ .

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Step 10.1

Convert the inequality to an equation.

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

Step 10.2

Use the quadratic formula to find the solutions.

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

Step 10.3

Substitute the values

$a = - 1$ ,

$b = 4$ , and

$c = - z + 4$ into the quadratic formula and solve for

$x$ .

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

Step 10.4

Simplify.

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Step 10.4.1

Simplify the numerator.

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Step 10.4.1.1

Raise

$4$ to the power of

$2$ .

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

Step 10.4.1.2

Multiply

$- 4$ by

$- 1$ .

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

Step 10.4.1.3

Apply the distributive property.

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

Step 10.4.1.4

Multiply

$- 1$ by

$4$ .

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

Step 10.4.1.5

Multiply

$4$ by

$4$ .

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

Step 10.4.1.6

Add

$16$ and

$16$ .

$x = \frac{- 4 \pm \sqrt{- 4 z + 32}}{2 \cdot - 1}$

Step 10.4.1.7

Factor

$4$ out of

$- 4 z + 32$ .

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Step 10.4.1.7.1

Factor

$4$ out of

$- 4 z$ .

$x = \frac{- 4 \pm \sqrt{4 (- z) + 32}}{2 \cdot - 1}$

Step 10.4.1.7.2

Factor

$4$ out of

$32$ .

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

Step 10.4.1.7.3

Factor

$4$ out of

$4 (- z) + 4 (8)$ .

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

Step 10.4.1.8

Rewrite

$4$ as

$2^{2}$ .

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

Step 10.4.1.9

Pull terms out from under the radical.

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

Step 10.4.2

Multiply

$2$ by

$- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$

Step 10.4.3

Simplify

$\frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$ .

$x = 2 \pm \sqrt{- z + 8}$

Step 10.5

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 10.5.1

Simplify the numerator.

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Step 10.5.1.1

Raise

$4$ to the power of

$2$ .

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

Step 10.5.1.2

Multiply

$- 4$ by

$- 1$ .

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

Step 10.5.1.3

Apply the distributive property.

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

Step 10.5.1.4

Multiply

$- 1$ by

$4$ .

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

Step 10.5.1.5

Multiply

$4$ by

$4$ .

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

Step 10.5.1.6

Add

$16$ and

$16$ .

$x = \frac{- 4 \pm \sqrt{- 4 z + 32}}{2 \cdot - 1}$

Step 10.5.1.7

Factor

$4$ out of

$- 4 z + 32$ .

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Step 10.5.1.7.1

Factor

$4$ out of

$- 4 z$ .

$x = \frac{- 4 \pm \sqrt{4 (- z) + 32}}{2 \cdot - 1}$

Step 10.5.1.7.2

Factor

$4$ out of

$32$ .

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

Step 10.5.1.7.3

Factor

$4$ out of

$4 (- z) + 4 (8)$ .

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

Step 10.5.1.8

Rewrite

$4$ as

$2^{2}$ .

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

Step 10.5.1.9

Pull terms out from under the radical.

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

Step 10.5.2

Multiply

$2$ by

$- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$

Step 10.5.3

Simplify

$\frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$ .

$x = 2 \pm \sqrt{- z + 8}$

Step 10.5.4

Change the

$\pm$ to

$+$ .

$x = 2 + \sqrt{- z + 8}$

Step 10.6

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 10.6.1

Simplify the numerator.

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Step 10.6.1.1

Raise

$4$ to the power of

$2$ .

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

Step 10.6.1.2

Multiply

$- 4$ by

$- 1$ .

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

Step 10.6.1.3

Apply the distributive property.

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

Step 10.6.1.4

Multiply

$- 1$ by

$4$ .

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

Step 10.6.1.5

Multiply

$4$ by

$4$ .

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

Step 10.6.1.6

Add

$16$ and

$16$ .

$x = \frac{- 4 \pm \sqrt{- 4 z + 32}}{2 \cdot - 1}$

Step 10.6.1.7

Factor

$4$ out of

$- 4 z + 32$ .

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Step 10.6.1.7.1

Factor

$4$ out of

$- 4 z$ .

$x = \frac{- 4 \pm \sqrt{4 (- z) + 32}}{2 \cdot - 1}$

Step 10.6.1.7.2

Factor

$4$ out of

$32$ .

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

Step 10.6.1.7.3

Factor

$4$ out of

$4 (- z) + 4 (8)$ .

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

Step 10.6.1.8

Rewrite

$4$ as

$2^{2}$ .

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

Step 10.6.1.9

Pull terms out from under the radical.

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

Step 10.6.2

Multiply

$2$ by

$- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$

Step 10.6.3

Simplify

$\frac{- 4 \pm 2 \sqrt{- z + 8}}{- 2}$ .

$x = 2 \pm \sqrt{- z + 8}$

Step 10.6.4

Change the

$\pm$ to

$-$ .

$x = 2 - \sqrt{- z + 8}$

Step 10.7

Consolidate the solutions.

$x = 2 + \sqrt{- z + 8}$

$x = 2 - \sqrt{- z + 8}$

$x = 2 + \sqrt{- z + 8}$

$x = 2 - \sqrt{- z + 8}$

Step 11

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$