Calculus Examples

$4 x^{2} + y^{2} + 4 z^{2} - 4 y - 24 z + 36 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $y^{2}$ from both sides of the equation.

$4 x^{2} + 4 z^{2} - 4 y - 24 z + 36 = - y^{2}$

Step 1.2

Subtract $4 z^{2}$ from both sides of the equation.

$4 x^{2} - 4 y - 24 z + 36 = - y^{2} - 4 z^{2}$

Step 1.3

Add $4 y$ to both sides of the equation.

$4 x^{2} - 24 z + 36 = - y^{2} - 4 z^{2} + 4 y$

Step 1.4

Add $24 z$ to both sides of the equation.

$4 x^{2} + 36 = - y^{2} - 4 z^{2} + 4 y + 24 z$

Step 1.5

Subtract $36$ from both sides of the equation.

$4 x^{2} = - y^{2} - 4 z^{2} + 4 y + 24 z - 36$

Step 2

Divide each term in $4 x^{2} = - y^{2} - 4 z^{2} + 4 y + 24 z - 36$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 x^{2} = - y^{2} - 4 z^{2} + 4 y + 24 z - 36$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{- y^{2}}{4} + \frac{- 4 z^{2}}{4} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{- y^{2}}{4} + \frac{- 4 z^{2}}{4} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.2.1.2

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

$x^{2} = \frac{- y^{2}}{4} + \frac{- 4 z^{2}}{4} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$x^{2} = - \frac{y^{2}}{4} + \frac{- 4 z^{2}}{4} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.2

Cancel the common factor of $- 4$ and $4$ .

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Step 2.3.1.2.1

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

$x^{2} = - \frac{y^{2}}{4} + \frac{4 (- z^{2})}{4} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.2.2

Cancel the common factors.

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Step 2.3.1.2.2.1

Factor $4$ out of $4$ .

$x^{2} = - \frac{y^{2}}{4} + \frac{4 (- z^{2})}{4 (1)} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.2.2.2

Cancel the common factor.

$x^{2} = - \frac{y^{2}}{4} + \frac{4 (- z^{2})}{4 \cdot 1} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.2.2.3

Rewrite the expression.

$x^{2} = - \frac{y^{2}}{4} + \frac{- z^{2}}{1} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.2.2.4

Divide $- z^{2}$ by $1$ .

$x^{2} = - \frac{y^{2}}{4} - z^{2} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.3

Cancel the common factor of $4$ .

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Step 2.3.1.3.1

Cancel the common factor.

$x^{2} = - \frac{y^{2}}{4} - z^{2} + \frac{4 y}{4} + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.3.2

Divide $y$ by $1$ .

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + \frac{24 z}{4} + \frac{- 36}{4}$

Step 2.3.1.4

Cancel the common factor of $24$ and $4$ .

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Step 2.3.1.4.1

Factor $4$ out of $24 z$ .

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

Step 2.3.1.4.2

Cancel the common factors.

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Step 2.3.1.4.2.1

Factor $4$ out of $4$ .

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + \frac{4 (6 z)}{4 (1)} + \frac{- 36}{4}$

Step 2.3.1.4.2.2

Cancel the common factor.

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + \frac{4 (6 z)}{4 \cdot 1} + \frac{- 36}{4}$

Step 2.3.1.4.2.3

Rewrite the expression.

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + \frac{6 z}{1} + \frac{- 36}{4}$

Step 2.3.1.4.2.4

Divide $6 z$ by $1$ .

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + 6 z + \frac{- 36}{4}$

Step 2.3.1.5

Divide $- 36$ by $4$ .

$x^{2} = - \frac{y^{2}}{4} - z^{2} + y + 6 z - 9$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{y^{2}}{4} - z^{2} + y + 6 z - 9}$

Step 4

Simplify $\pm \sqrt{- \frac{y^{2}}{4} - z^{2} + y + 6 z - 9}$ .

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Step 4.1

To write $- z^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{- \frac{y^{2}}{4} - z^{2} \cdot \frac{4}{4} + y + 6 z - 9}$

Step 4.2

Combine $- z^{2}$ and $\frac{4}{4}$ .

$x = \pm \sqrt{- \frac{y^{2}}{4} + \frac{- z^{2} \cdot 4}{4} + y + 6 z - 9}$

Step 4.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- y^{2} - z^{2} \cdot 4}{4} + y + 6 z - 9}$

Step 4.4

Multiply $4$ by $- 1$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2}}{4} + y + 6 z - 9}$

Step 4.5

To write $y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2}}{4} + y \cdot \frac{4}{4} + 6 z - 9}$

Step 4.6

Simplify terms.

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Step 4.6.1

Combine $y$ and $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2}}{4} + \frac{y \cdot 4}{4} + 6 z - 9}$

Step 4.6.2

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + y \cdot 4}{4} + 6 z - 9}$

Step 4.7

Move $4$ to the left of $y$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y}{4} + 6 z - 9}$

Step 4.8

To write $6 z$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y}{4} + 6 z \cdot \frac{4}{4} - 9}$

Step 4.9

Combine $6 z$ and $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y}{4} + \frac{6 z \cdot 4}{4} - 9}$

Step 4.10

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 6 z \cdot 4}{4} - 9}$

Step 4.11

Multiply $4$ by $6$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 24 z}{4} - 9}$

Step 4.12

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 24 z}{4} - 9 \cdot \frac{4}{4}}$

Step 4.13

Combine $- 9$ and $\frac{4}{4}$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 24 z}{4} + \frac{- 9 \cdot 4}{4}}$

Step 4.14

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 24 z - 9 \cdot 4}{4}}$

Step 4.15

Multiply $- 9$ by $4$ .

$x = \pm \sqrt{\frac{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}{4}}$

Step 4.16

Rewrite $\frac{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}{4}$ as ${(\frac{1}{2})}^{2} (- y^{2} - 4 z^{2} + 4 y + 24 z - 36)$ .

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Step 4.16.1

Factor the perfect power $1^{2}$ out of $- y^{2} - 4 z^{2} + 4 y + 24 z - 36$ .

$x = \pm \sqrt{\frac{1^{2} (- y^{2} - 4 z^{2} + 4 y + 24 z - 36)}{4}}$

Step 4.16.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} (- y^{2} - 4 z^{2} + 4 y + 24 z - 36)}{2^{2} \cdot 1}}$

Step 4.16.3

Rearrange the fraction $\frac{1^{2} (- y^{2} - 4 z^{2} + 4 y + 24 z - 36)}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} (- y^{2} - 4 z^{2} + 4 y + 24 z - 36)}$

Step 4.17

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}$

Step 4.18

Combine $\frac{1}{2}$ and $\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}$ .

$x = \pm \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

$x = - \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

$x = \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$

$x = - \frac{\sqrt{- y^{2} - 4 z^{2} + 4 y + 24 z - 36}}{2}$