Calculus Examples

$2 x^{2} + 2 y^{2} + 2 z^{2} + x + y + z = 9$

Step 1

Complete the square for $2 x^{2} + x$ .

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Step 1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 2$

$b = 1$

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot 2}$

Step 1.3.2

Multiply $2$ by $2$ .

$d = \frac{1}{4}$

Step 1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot 2}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

One to any power is one.

$e = 0 - \frac{1}{4 \cdot 2}$

Step 1.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{1}{8}$

Step 1.4.2.2

Subtract $\frac{1}{8}$ from $0$ .

$e = - \frac{1}{8}$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{1}{4})}^{2} - \frac{1}{8}$ .

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

Step 2

Substitute $2 {(x + \frac{1}{4})}^{2} - \frac{1}{8}$ for $2 x^{2} + x$ in the equation $2 x^{2} + 2 y^{2} + 2 z^{2} + x + y + z = 9$ .

$2 {(x + \frac{1}{4})}^{2} - \frac{1}{8} + 2 y^{2} + 2 z^{2} + y + z = 9$

Step 3

Move $- \frac{1}{8}$ to the right side of the equation by adding $\frac{1}{8}$ to both sides.

$2 {(x + \frac{1}{4})}^{2} + 2 y^{2} + 2 z^{2} + y + z = 9 + \frac{1}{8}$

Step 4

Complete the square for $2 y^{2} + y$ .

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Step 4.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 2$

$b = 1$

$c = 0$

Step 4.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot 2}$

Step 4.3.2

Multiply $2$ by $2$ .

$d = \frac{1}{4}$

Step 4.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 4.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot 2}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

One to any power is one.

$e = 0 - \frac{1}{4 \cdot 2}$

Step 4.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{1}{8}$

Step 4.4.2.2

Subtract $\frac{1}{8}$ from $0$ .

$e = - \frac{1}{8}$

Step 4.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(y + \frac{1}{4})}^{2} - \frac{1}{8}$ .

$2 {(y + \frac{1}{4})}^{2} - \frac{1}{8}$

Step 5

Substitute $2 {(y + \frac{1}{4})}^{2} - \frac{1}{8}$ for $2 y^{2} + y$ in the equation $2 x^{2} + 2 y^{2} + 2 z^{2} + x + y + z = 9$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} - \frac{1}{8} = 9 + \frac{1}{8}$

Step 6

Move $- \frac{1}{8}$ to the right side of the equation by adding $\frac{1}{8}$ to both sides.

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} = 9 + \frac{1}{8} + \frac{1}{8}$

Step 7

Complete the square for $2 z^{2} + z$ .

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Step 7.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 2$

$b = 1$

$c = 0$

Step 7.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 7.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 7.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{1}{2 \cdot 2}$

Step 7.3.2

Multiply $2$ by $2$ .

$d = \frac{1}{4}$

Step 7.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 7.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{1^{2}}{4 \cdot 2}$

Step 7.4.2

Simplify the right side.

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Step 7.4.2.1

Simplify each term.

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Step 7.4.2.1.1

One to any power is one.

$e = 0 - \frac{1}{4 \cdot 2}$

Step 7.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{1}{8}$

Step 7.4.2.2

Subtract $\frac{1}{8}$ from $0$ .

$e = - \frac{1}{8}$

Step 7.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(z + \frac{1}{4})}^{2} - \frac{1}{8}$ .

$2 {(z + \frac{1}{4})}^{2} - \frac{1}{8}$

Step 8

Substitute $2 {(z + \frac{1}{4})}^{2} - \frac{1}{8}$ for $2 z^{2} + z$ in the equation $2 x^{2} + 2 y^{2} + 2 z^{2} + x + y + z = 9$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} - \frac{1}{8} = 9 + \frac{1}{8} + \frac{1}{8}$

Step 9

Move $- \frac{1}{8}$ to the right side of the equation by adding $\frac{1}{8}$ to both sides.

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = 9 + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$

Step 10

Simplify $9 + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}$ .

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Step 10.1

Combine the numerators over the common denominator.

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = 9 + \frac{1 + 1 + 1}{8}$

Step 10.2

Simplify by adding numbers.

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Step 10.2.1

Add $1$ and $1$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = 9 + \frac{2 + 1}{8}$

Step 10.2.2

Add $2$ and $1$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = 9 + \frac{3}{8}$

Step 10.3

To write $9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = 9 \cdot \frac{8}{8} + \frac{3}{8}$

Step 10.4

Combine $9$ and $\frac{8}{8}$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = \frac{9 \cdot 8}{8} + \frac{3}{8}$

Step 10.5

Combine the numerators over the common denominator.

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = \frac{9 \cdot 8 + 3}{8}$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $9$ by $8$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = \frac{72 + 3}{8}$

Step 10.6.2

Add $72$ and $3$ .

$2 {(x + \frac{1}{4})}^{2} + 2 {(y + \frac{1}{4})}^{2} + 2 {(z + \frac{1}{4})}^{2} = \frac{75}{8}$

Step 11