Precalculus Examples

$2 x^{2} - 2 x + 2 y^{2} - 6 y + 2 z^{2} - 4 z + 5$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 2$ , $b = - 6$ , and $c = 2 x^{2} - 2 x + 2 z^{2} - 4 z + 5$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (2 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5))}}{2 \cdot 2}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 3.1.2

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 3.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 8 (2 x^{2}) - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 3.1.4.2

Multiply $- 2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 3.1.4.3

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 3.1.4.4

Multiply $- 4$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 8 \cdot 5}}{2 \cdot 2}$

Step 3.1.4.5

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 40}}{2 \cdot 2}$

Step 3.1.5

Subtract $40$ from $36$ .

$y = \frac{6 \pm \sqrt{- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 3.1.6

Factor $4$ out of $- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4$ .

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Step 3.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 3.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 3.1.6.3

Factor $4$ out of $- 16 z^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 32 z - 4}}{2 \cdot 2}$

Step 3.1.6.4

Factor $4$ out of $32 z$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) - 4}}{2 \cdot 2}$

Step 3.1.6.5

Factor $4$ out of $- 4$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 3.1.6.6

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 3.1.6.7

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 3.1.6.8

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 3.1.6.9

Factor $4$ out of $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 3.1.7

Rewrite $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)$ as $2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)$ .

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Step 3.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 3.1.7.2

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 3.1.9

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

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$

Step 3.3

Simplify $\frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$ .

$y = \frac{3 \pm \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 4.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 8 (2 x^{2}) - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.4.2

Multiply $- 2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.4.3

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.4.4

Multiply $- 4$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 8 \cdot 5}}{2 \cdot 2}$

Step 4.1.4.5

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 40}}{2 \cdot 2}$

Step 4.1.5

Subtract $40$ from $36$ .

$y = \frac{6 \pm \sqrt{- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 4.1.6

Factor $4$ out of $- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4$ .

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Step 4.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 4.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 4.1.6.3

Factor $4$ out of $- 16 z^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 32 z - 4}}{2 \cdot 2}$

Step 4.1.6.4

Factor $4$ out of $32 z$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) - 4}}{2 \cdot 2}$

Step 4.1.6.5

Factor $4$ out of $- 4$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 4.1.6.6

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 4.1.6.7

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 4.1.6.8

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 4.1.6.9

Factor $4$ out of $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 4.1.7

Rewrite $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)$ as $2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)$ .

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Step 4.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 4.1.7.2

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 4.1.9

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

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$ .

$y = \frac{3 \pm \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 4.4

Change the $\pm$ to $+$ .

$y = \frac{3 + \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{6 \pm \sqrt{36 - 8 \cdot (2 x^{2} - 2 x + 2 z^{2} - 4 z + 5)}}{2 \cdot 2}$

Step 5.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 8 (2 x^{2}) - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} - 8 (- 2 x) - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.4.2

Multiply $- 2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 8 (2 z^{2}) - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.4.3

Multiply $2$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} - 8 (- 4 z) - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.4.4

Multiply $- 4$ by $- 8$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 8 \cdot 5}}{2 \cdot 2}$

Step 5.1.4.5

Multiply $- 8$ by $5$ .

$y = \frac{6 \pm \sqrt{36 - 16 x^{2} + 16 x - 16 z^{2} + 32 z - 40}}{2 \cdot 2}$

Step 5.1.5

Subtract $40$ from $36$ .

$y = \frac{6 \pm \sqrt{- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 5.1.6

Factor $4$ out of $- 16 x^{2} + 16 x - 16 z^{2} + 32 z - 4$ .

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Step 5.1.6.1

Factor $4$ out of $- 16 x^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 16 x - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 5.1.6.2

Factor $4$ out of $16 x$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) - 16 z^{2} + 32 z - 4}}{2 \cdot 2}$

Step 5.1.6.3

Factor $4$ out of $- 16 z^{2}$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 32 z - 4}}{2 \cdot 2}$

Step 5.1.6.4

Factor $4$ out of $32 z$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) - 4}}{2 \cdot 2}$

Step 5.1.6.5

Factor $4$ out of $- 4$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2}) + 4 (4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 5.1.6.6

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x) + 4 (- 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 5.1.6.7

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2}) + 4 (8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 5.1.6.8

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

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)}}{2 \cdot 2}$

Step 5.1.6.9

Factor $4$ out of $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z) + 4 (- 1)$ .

$y = \frac{6 \pm \sqrt{4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 5.1.7

Rewrite $4 (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)$ as $2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)$ .

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Step 5.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 5.1.7.2

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

$y = \frac{6 \pm \sqrt{2^{2} (- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1)}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 2^{2} x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 5.1.9

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

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$

Step 5.3

Simplify $\frac{6 \pm 2 \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{4}$ .

$y = \frac{3 \pm \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 5.4

Change the $\pm$ to $-$ .

$y = \frac{3 - \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{3 + \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

$y = \frac{3 - \sqrt{- 4 x^{2} + 4 x - 4 z^{2} + 8 z - 1}}{2}$

Step 7