Linear Algebra Examples

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

Step 1

Rewrite the equation as $4 x - 4 y - x^{2} - y^{2} = z$ .

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

Step 2

Subtract $z$ from both sides of the equation.

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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Step 5.1.1.1

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

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

Step 5.1.1.2

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

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

Step 5.1.1.3

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

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

Step 5.1.2

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

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Step 5.1.2.1

Reorder the expression.

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Step 5.1.2.1.1

Move $4 x$ .

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

Step 5.1.2.1.2

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

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

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

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

Step 5.1.2.4

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

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

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

Step 5.1.3

Combine exponents.

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Step 5.1.3.1

Factor out negative.

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

Step 5.1.3.2

Multiply $- 4$ by $- 1$ .

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

Step 5.1.4

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

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Step 5.1.4.1

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

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

Step 5.1.4.2

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

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

Step 5.1.5

Pull terms out from under the radical.

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

Step 5.1.6

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

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}$ .

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

Step 5.5

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

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

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

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

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

Step 6.1.1.3

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

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

Step 6.1.2

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

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Step 6.1.2.1

Reorder the expression.

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Step 6.1.2.1.1

Move $4 x$ .

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

Step 6.1.2.1.2

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

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

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

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

Step 6.1.2.4

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

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

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

Step 6.1.3

Combine exponents.

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Step 6.1.3.1

Factor out negative.

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

Step 6.1.3.2

Multiply $- 4$ by $- 1$ .

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

Step 6.1.4

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

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Step 6.1.4.1

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

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

Step 6.1.4.2

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

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

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.1.6

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

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

Step 6.2

Multiply $2$ by $- 1$ .

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

Step 6.3

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

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

Step 6.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{- x^{2} - z + 4 x + 4}}{- 1}$ .

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

Step 6.5

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

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

Step 6.6

Change the $\pm$ to $+$ .

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

Step 6.7

Apply the distributive property.

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

Step 6.8

Multiply $- 1$ by $2$ .

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

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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Step 7.1.1.1

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

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

Step 7.1.1.2

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

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

Step 7.1.1.3

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

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

Step 7.1.2

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

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Step 7.1.2.1

Reorder the expression.

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Step 7.1.2.1.1

Move $4 x$ .

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

Step 7.1.2.1.2

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

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

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

Step 7.1.2.2

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

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

Step 7.1.2.3

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

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

Step 7.1.2.4

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

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

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

Step 7.1.3

Combine exponents.

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Step 7.1.3.1

Factor out negative.

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

Step 7.1.3.2

Multiply $- 4$ by $- 1$ .

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

Step 7.1.4

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

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Step 7.1.4.1

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

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

Step 7.1.4.2

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

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

Step 7.1.5

Pull terms out from under the radical.

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

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:

${x | x \in ℝ}$