Precalculus Examples

$x^{2} - 7 x y + 6 y^{2} = 0$ , $x^{2} + x y = 42$

Step 1

Solve for $y$ in $x^{2} + x y = 42$ .

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Step 1.1

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

$x y = 42 - x^{2}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2

Divide each term in $x y = 42 - x^{2}$ by $x$ and simplify.

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Step 1.2.1

Divide each term in $x y = 42 - x^{2}$ by $x$ .

$\frac{x y}{x} = \frac{42}{x} + \frac{- x^{2}}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $x$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{42}{x} + \frac{- x^{2}}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{42}{x} + \frac{- x^{2}}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} + \frac{- x^{2}}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} + \frac{- x^{2}}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.3.1.1

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

$y = \frac{42}{x} + \frac{x (- x)}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{42}{x} + \frac{x (- x)}{x}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3.1.2.2

Factor $x$ out of $x^{1}$ .

$y = \frac{42}{x} + \frac{x (- x)}{x \cdot 1}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3.1.2.3

Cancel the common factor.

$y = \frac{42}{x} + \frac{x (- x)}{x \cdot 1}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3.1.2.4

Rewrite the expression.

$y = \frac{42}{x} + \frac{- x}{1}$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 1.2.3.1.2.5

Divide $- x$ by $1$ .

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 x y + 6 y^{2} = 0$

Step 2

Replace all occurrences of $y$ with $\frac{42}{x} - x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} - 7 x y + 6 y^{2} = 0$ with $\frac{42}{x} - x$ .

$x^{2} - 7 x (\frac{42}{x} - x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} - 7 x (\frac{42}{x} - x) + 6 {(\frac{42}{x} - x)}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$x^{2} - 7 x \frac{42}{x} - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.1.2.1

Factor $x$ out of $- 7 x$ .

$x^{2} + x \cdot (- 7 \frac{42}{x}) - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.2.2

Cancel the common factor.

$x^{2} + x \cdot (- 7 \frac{42}{x}) - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.2.3

Rewrite the expression.

$x^{2} - 7 \cdot 42 - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 7 \cdot 42 - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.3

Multiply $- 7$ by $42$ .

$x^{2} - 294 - 7 x (- x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.4

Rewrite using the commutative property of multiplication.

$x^{2} - 294 - 7 \cdot (- 1 x \cdot x) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.5

Simplify each term.

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Step 2.2.1.1.5.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.1.5.1.1

Move $x$ .

$x^{2} - 294 - 7 \cdot (- 1 (x \cdot x)) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.5.1.2

Multiply $x$ by $x$ .

$x^{2} - 294 - 7 \cdot (- 1 x^{2}) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 - 7 \cdot (- 1 x^{2}) + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.5.2

Multiply $- 7$ by $- 1$ .

$x^{2} - 294 + 7 x^{2} + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 {(\frac{42}{x} - x)}^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.6

Rewrite ${(\frac{42}{x} - x)}^{2}$ as $(\frac{42}{x} - x) (\frac{42}{x} - x)$ .

$x^{2} - 294 + 7 x^{2} + 6 ((\frac{42}{x} - x) (\frac{42}{x} - x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.7

Expand $(\frac{42}{x} - x) (\frac{42}{x} - x)$ using the FOIL Method.

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Step 2.2.1.1.7.1

Apply the distributive property.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{42}{x} \cdot (\frac{42}{x} - x) - x (\frac{42}{x} - x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.7.2

Apply the distributive property.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{42}{x} \cdot \frac{42}{x} + \frac{42}{x} \cdot (- x) - x (\frac{42}{x} - x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.7.3

Apply the distributive property.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{42}{x} \cdot \frac{42}{x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{42}{x} \cdot \frac{42}{x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8

Simplify and combine like terms.

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Step 2.2.1.1.8.1

Simplify each term.

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Step 2.2.1.1.8.1.1

Multiply $\frac{42}{x} \cdot \frac{42}{x}$ .

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Step 2.2.1.1.8.1.1.1

Multiply $\frac{42}{x}$ by $\frac{42}{x}$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{42 \cdot 42}{x \cdot x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.1.2

Multiply $42$ by $42$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x \cdot x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.1.3

Raise $x$ to the power of $1$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x \cdot x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.1.4

Raise $x$ to the power of $1$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x \cdot x} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{1 + 1}} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.1.6

Add $1$ and $1$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} + \frac{42}{x} \cdot (- x) - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.2

Rewrite using the commutative property of multiplication.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - \frac{42}{x} \cdot x - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.3

Cancel the common factor of $x$ .

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Step 2.2.1.1.8.1.3.1

Move the leading negative in $- \frac{42}{x}$ into the numerator.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} + \frac{- 42}{x} \cdot x - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.3.2

Cancel the common factor.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} + \frac{- 42}{x} \cdot x - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.3.3

Rewrite the expression.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - x \frac{42}{x} - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.4

Cancel the common factor of $x$ .

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Step 2.2.1.1.8.1.4.1

Factor $x$ out of $- x$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 + x \cdot (- 1 \frac{42}{x}) - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.4.2

Cancel the common factor.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 + x \cdot (- 1 \frac{42}{x}) - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.4.3

Rewrite the expression.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 1 \cdot 42 - x (- x)) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 1 \cdot 42 - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.5

Multiply $- 1$ by $42$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 - x (- x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.6

Rewrite using the commutative property of multiplication.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 - 1 \cdot (- 1 x \cdot x)) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.7

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.1.8.1.7.1

Move $x$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 - 1 \cdot (- 1 (x \cdot x))) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.7.2

Multiply $x$ by $x$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 - 1 \cdot (- 1 x^{2})) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 - 1 \cdot (- 1 x^{2})) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.8

Multiply $- 1$ by $- 1$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 + 1 x^{2}) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.1.9

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

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 + x^{2}) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 42 - 42 + x^{2}) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.8.2

Subtract $42$ from $- 42$ .

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 84 + x^{2}) = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}} - 84 + x^{2}) = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.9

Apply the distributive property.

$x^{2} - 294 + 7 x^{2} + 6 (\frac{1764}{x^{2}}) + 6 \cdot - 84 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.10

Simplify.

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Step 2.2.1.1.10.1

Multiply $6 \frac{1764}{x^{2}}$ .

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Step 2.2.1.1.10.1.1

Combine $6$ and $\frac{1764}{x^{2}}$ .

$x^{2} - 294 + 7 x^{2} + \frac{6 \cdot 1764}{x^{2}} + 6 \cdot - 84 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.10.1.2

Multiply $6$ by $1764$ .

$x^{2} - 294 + 7 x^{2} + \frac{10584}{x^{2}} + 6 \cdot - 84 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + \frac{10584}{x^{2}} + 6 \cdot - 84 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.1.10.2

Multiply $6$ by $- 84$ .

$x^{2} - 294 + 7 x^{2} + \frac{10584}{x^{2}} - 504 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + \frac{10584}{x^{2}} - 504 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

$x^{2} - 294 + 7 x^{2} + \frac{10584}{x^{2}} - 504 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x^{2}$ and $7 x^{2}$ .

$8 x^{2} - 294 + \frac{10584}{x^{2}} - 504 + 6 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.2.2

Add $8 x^{2}$ and $6 x^{2}$ .

$14 x^{2} - 294 + \frac{10584}{x^{2}} - 504 = 0$

$y = \frac{42}{x} - x$

Step 2.2.1.2.3

Subtract $504$ from $- 294$ .

$14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$

$y = \frac{42}{x} - x$

$14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$

$y = \frac{42}{x} - x$

$14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$

$y = \frac{42}{x} - x$

$14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$

$y = \frac{42}{x} - x$

$14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$

$y = \frac{42}{x} - x$

Step 3

Solve for $x$ in $14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1, 1$

$y = \frac{42}{x} - x$

Step 3.1.2

The LCM of one and any expression is the expression.

$x^{2}$

$y = \frac{42}{x} - x$

$x^{2}$

$y = \frac{42}{x} - x$

Step 3.2

Multiply each term in $14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$ by $x^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $14 x^{2} + \frac{10584}{x^{2}} - 798 = 0$ by $x^{2}$ .

$14 x^{2} x^{2} + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.2.2.1.1.1

Move $x^{2}$ .

$14 (x^{2} x^{2}) + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$14 x^{2 + 2} + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.2.1.1.3

Add $2$ and $2$ .

$14 x^{4} + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

$14 x^{4} + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.2.1.2.1

Cancel the common factor.

$14 x^{4} + \frac{10584}{x^{2}} \cdot x^{2} - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.2.1.2.2

Rewrite the expression.

$14 x^{4} + 10584 - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

$14 x^{4} + 10584 - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

$14 x^{4} + 10584 - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

$14 x^{4} + 10584 - 798 x^{2} = 0 x^{2}$

$y = \frac{42}{x} - x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply $0$ by $x^{2}$ .

$14 x^{4} + 10584 - 798 x^{2} = 0$

$y = \frac{42}{x} - x$

$14 x^{4} + 10584 - 798 x^{2} = 0$

$y = \frac{42}{x} - x$

$14 x^{4} + 10584 - 798 x^{2} = 0$

$y = \frac{42}{x} - x$

Step 3.3

Solve the equation.

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Step 3.3.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$14 u^{2} - 798 u + 10584 = 0$

$u = x^{2}$

$y = \frac{42}{x} - x$

Step 3.3.2

Factor the left side of the equation.

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Step 3.3.2.1

Factor $14$ out of $14 u^{2} - 798 u + 10584$ .

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Step 3.3.2.1.1

Factor $14$ out of $14 u^{2}$ .

$14 (u^{2}) - 798 u + 10584 = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.1.2

Factor $14$ out of $- 798 u$ .

$14 (u^{2}) + 14 (- 57 u) + 10584 = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.1.3

Factor $14$ out of $10584$ .

$14 u^{2} + 14 (- 57 u) + 14 \cdot 756 = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.1.4

Factor $14$ out of $14 u^{2} + 14 (- 57 u)$ .

$14 (u^{2} - 57 u) + 14 \cdot 756 = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.1.5

Factor $14$ out of $14 (u^{2} - 57 u) + 14 \cdot 756$ .

$14 (u^{2} - 57 u + 756) = 0$

$y = \frac{42}{x} - x$

$14 (u^{2} - 57 u + 756) = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.2

Factor.

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Step 3.3.2.2.1

Factor $u^{2} - 57 u + 756$ using the AC method.

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Step 3.3.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $756$ and whose sum is $- 57$ .

$- 36, - 21$

$y = \frac{42}{x} - x$

Step 3.3.2.2.1.2

Write the factored form using these integers.

$14 ((u - 36) (u - 21)) = 0$

$y = \frac{42}{x} - x$

$14 ((u - 36) (u - 21)) = 0$

$y = \frac{42}{x} - x$

Step 3.3.2.2.2

Remove unnecessary parentheses.

$14 (u - 36) (u - 21) = 0$

$y = \frac{42}{x} - x$

$14 (u - 36) (u - 21) = 0$

$y = \frac{42}{x} - x$

$14 (u - 36) (u - 21) = 0$

$y = \frac{42}{x} - x$

Step 3.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u - 36 = 0$

$u - 21 = 0$

$y = \frac{42}{x} - x$

Step 3.3.4

Set $u - 36$ equal to $0$ and solve for $u$ .

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Step 3.3.4.1

Set $u - 36$ equal to $0$ .

$u - 36 = 0$

$y = \frac{42}{x} - x$

Step 3.3.4.2

Add $36$ to both sides of the equation.

$u = 36$

$y = \frac{42}{x} - x$

$u = 36$

$y = \frac{42}{x} - x$

Step 3.3.5

Set $u - 21$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $u - 21$ equal to $0$ .

$u - 21 = 0$

$y = \frac{42}{x} - x$

Step 3.3.5.2

Add $21$ to both sides of the equation.

$u = 21$

$y = \frac{42}{x} - x$

$u = 21$

$y = \frac{42}{x} - x$

Step 3.3.6

The final solution is all the values that make $14 (u - 36) (u - 21) = 0$ true.

$u = 36, 21$

$y = \frac{42}{x} - x$

Step 3.3.7

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 36$

$x^{2} = 21$

$y = \frac{42}{x} - x$

Step 3.3.8

Solve the first equation for $x$ .

$x^{2} = 36$

$y = \frac{42}{x} - x$

Step 3.3.9

Solve the equation for $x$ .

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Step 3.3.9.1

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

$x = \pm \sqrt{36}$

$y = \frac{42}{x} - x$

Step 3.3.9.2

Simplify $\pm \sqrt{36}$ .

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Step 3.3.9.2.1

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

$y = \frac{42}{x} - x$

Step 3.3.9.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 6$

$y = \frac{42}{x} - x$

$x = \pm 6$

$y = \frac{42}{x} - x$

Step 3.3.9.3

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

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Step 3.3.9.3.1

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

$x = 6$

$y = \frac{42}{x} - x$

Step 3.3.9.3.2

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

$x = - 6$

$y = \frac{42}{x} - x$

Step 3.3.9.3.3

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

$x = 6, - 6$

$y = \frac{42}{x} - x$

$x = 6, - 6$

$y = \frac{42}{x} - x$

$x = 6, - 6$

$y = \frac{42}{x} - x$

Step 3.3.10

Solve the second equation for $x$ .

$x^{2} = 21$

$y = \frac{42}{x} - x$

Step 3.3.11

Solve the equation for $x$ .

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Step 3.3.11.1

Remove parentheses.

$x^{2} = 21$

$y = \frac{42}{x} - x$

Step 3.3.11.2

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

$x = \pm \sqrt{21}$

$y = \frac{42}{x} - x$

Step 3.3.11.3

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

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Step 3.3.11.3.1

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

$x = \sqrt{21}$

$y = \frac{42}{x} - x$

Step 3.3.11.3.2

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

$x = - \sqrt{21}$

$y = \frac{42}{x} - x$

Step 3.3.11.3.3

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

$x = \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

$x = \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

$x = \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

Step 3.3.12

The solution to $14 x^{4} - 798 x^{2} + 10584 = 0$ is $x = 6, - 6, \sqrt{21}, - \sqrt{21}$ .

$x = 6, - 6, \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

$x = 6, - 6, \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

$x = 6, - 6, \sqrt{21}, - \sqrt{21}$

$y = \frac{42}{x} - x$

Step 4

Replace all occurrences of $x$ with $6$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $6$ .

$y = \frac{42}{6} - (6)$

$x = 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{42}{6} - (6)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Divide $42$ by $6$ .

$y = 7 - (6)$

$x = 6$

Step 4.2.1.1.2

Multiply $- 1$ by $6$ .

$y = 7 - 6$

$x = 6$

$y = 7 - 6$

$x = 6$

Step 4.2.1.2

Subtract $6$ from $7$ .

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

Step 5

Replace all occurrences of $x$ with $- 6$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $- 6$ .

$y = \frac{42}{- 6} - (- 6)$

$x = - 6$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{42}{- 6} - (- 6)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Divide $42$ by $- 6$ .

$y = - 7 - (- 6)$

$x = - 6$

Step 5.2.1.1.2

Multiply $- 1$ by $- 6$ .

$y = - 7 + 6$

$x = - 6$

$y = - 7 + 6$

$x = - 6$

Step 5.2.1.2

Add $- 7$ and $6$ .

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

Step 6

Replace all occurrences of $x$ with $6$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $6$ .

$y = \frac{42}{6} - (6)$

$x = 6$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{42}{6} - (6)$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Divide $42$ by $6$ .

$y = 7 - (6)$

$x = 6$

Step 6.2.1.1.2

Multiply $- 1$ by $6$ .

$y = 7 - 6$

$x = 6$

$y = 7 - 6$

$x = 6$

Step 6.2.1.2

Subtract $6$ from $7$ .

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

Step 7

Replace all occurrences of $x$ with $- 6$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $- 6$ .

$y = \frac{42}{- 6} - (- 6)$

$x = - 6$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{42}{- 6} - (- 6)$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Divide $42$ by $- 6$ .

$y = - 7 - (- 6)$

$x = - 6$

Step 7.2.1.1.2

Multiply $- 1$ by $- 6$ .

$y = - 7 + 6$

$x = - 6$

$y = - 7 + 6$

$x = - 6$

Step 7.2.1.2

Add $- 7$ and $6$ .

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

Step 8

Replace all occurrences of $x$ with $\sqrt{21}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $\sqrt{21}$ .

$y = \frac{42}{\sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{42}{\sqrt{21}} - (\sqrt{21})$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \frac{42}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2

Combine and simplify the denominator.

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Step 8.2.1.1.2.1

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.2

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.3

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{42 \sqrt{21}}{{\sqrt{21}}^{1 + 1}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.5

Add $1$ and $1$ .

$y = \frac{42 \sqrt{21}}{{\sqrt{21}}^{2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 8.2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \frac{42 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{42 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 8.2.1.1.2.6.4.1

Cancel the common factor.

$y = \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6.4.2

Rewrite the expression.

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.2.6.5

Evaluate the exponent.

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.3

Cancel the common factor of $42$ and $21$ .

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Step 8.2.1.1.3.1

Factor $21$ out of $42 \sqrt{21}$ .

$y = \frac{21 (2 \sqrt{21})}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.3.2

Cancel the common factors.

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Step 8.2.1.1.3.2.1

Factor $21$ out of $21$ .

$y = \frac{21 (2 \sqrt{21})}{21 (1)} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.3.2.2

Cancel the common factor.

$y = \frac{21 (2 \sqrt{21})}{21 \cdot 1} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.3.2.3

Rewrite the expression.

$y = \frac{2 \sqrt{21}}{1} - (\sqrt{21})$

$x = \sqrt{21}$

Step 8.2.1.1.3.2.4

Divide $2 \sqrt{21}$ by $1$ .

$y = 2 \sqrt{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - \sqrt{21}$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - \sqrt{21}$

$x = \sqrt{21}$

Step 8.2.1.2

Subtract $\sqrt{21}$ from $2 \sqrt{21}$ .

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

Step 9

Replace all occurrences of $x$ with $6$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $6$ .

$y = \frac{42}{6} - (6)$

$x = 6$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{42}{6} - (6)$ .

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Step 9.2.1.1

Simplify each term.

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Step 9.2.1.1.1

Divide $42$ by $6$ .

$y = 7 - (6)$

$x = 6$

Step 9.2.1.1.2

Multiply $- 1$ by $6$ .

$y = 7 - 6$

$x = 6$

$y = 7 - 6$

$x = 6$

Step 9.2.1.2

Subtract $6$ from $7$ .

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

Step 10

Replace all occurrences of $x$ with $- 6$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $- 6$ .

$y = \frac{42}{- 6} - (- 6)$

$x = - 6$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $\frac{42}{- 6} - (- 6)$ .

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Step 10.2.1.1

Simplify each term.

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Step 10.2.1.1.1

Divide $42$ by $- 6$ .

$y = - 7 - (- 6)$

$x = - 6$

Step 10.2.1.1.2

Multiply $- 1$ by $- 6$ .

$y = - 7 + 6$

$x = - 6$

$y = - 7 + 6$

$x = - 6$

Step 10.2.1.2

Add $- 7$ and $6$ .

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

$y = - 1$

$x = - 6$

Step 11

Replace all occurrences of $x$ with $\sqrt{21}$ in each equation.

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Step 11.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $\sqrt{21}$ .

$y = \frac{42}{\sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{42}{\sqrt{21}} - (\sqrt{21})$ .

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Step 11.2.1.1

Simplify each term.

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Step 11.2.1.1.1

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \frac{42}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2

Combine and simplify the denominator.

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Step 11.2.1.1.2.1

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.2

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.3

Raise $\sqrt{21}$ to the power of $1$ .

$y = \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{42 \sqrt{21}}{{\sqrt{21}}^{1 + 1}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.5

Add $1$ and $1$ .

$y = \frac{42 \sqrt{21}}{{\sqrt{21}}^{2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 11.2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = \frac{42 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{42 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 11.2.1.1.2.6.4.1

Cancel the common factor.

$y = \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6.4.2

Rewrite the expression.

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.2.6.5

Evaluate the exponent.

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = \frac{42 \sqrt{21}}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.3

Cancel the common factor of $42$ and $21$ .

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Step 11.2.1.1.3.1

Factor $21$ out of $42 \sqrt{21}$ .

$y = \frac{21 (2 \sqrt{21})}{21} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.3.2

Cancel the common factors.

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Step 11.2.1.1.3.2.1

Factor $21$ out of $21$ .

$y = \frac{21 (2 \sqrt{21})}{21 (1)} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.3.2.2

Cancel the common factor.

$y = \frac{21 (2 \sqrt{21})}{21 \cdot 1} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.3.2.3

Rewrite the expression.

$y = \frac{2 \sqrt{21}}{1} - (\sqrt{21})$

$x = \sqrt{21}$

Step 11.2.1.1.3.2.4

Divide $2 \sqrt{21}$ by $1$ .

$y = 2 \sqrt{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - (\sqrt{21})$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - \sqrt{21}$

$x = \sqrt{21}$

$y = 2 \sqrt{21} - \sqrt{21}$

$x = \sqrt{21}$

Step 11.2.1.2

Subtract $\sqrt{21}$ from $2 \sqrt{21}$ .

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

$y = \sqrt{21}$

$x = \sqrt{21}$

Step 12

Replace all occurrences of $x$ with $- \sqrt{21}$ in each equation.

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Step 12.1

Replace all occurrences of $x$ in $y = \frac{42}{x} - x$ with $- \sqrt{21}$ .

$y = \frac{42}{- \sqrt{21}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{42}{- \sqrt{21}} - (- \sqrt{21})$ .

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Step 12.2.1.1

Simplify each term.

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Step 12.2.1.1.1

Move the negative in front of the fraction.

$y = - \frac{42}{\sqrt{21}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.2

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = - (\frac{42}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}) - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3

Combine and simplify the denominator.

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Step 12.2.1.1.3.1

Multiply $\frac{42}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$y = - \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.2

Raise $\sqrt{21}$ to the power of $1$ .

$y = - \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.3

Raise $\sqrt{21}$ to the power of $1$ .

$y = - \frac{42 \sqrt{21}}{\sqrt{21} \sqrt{21}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = - \frac{42 \sqrt{21}}{{\sqrt{21}}^{1 + 1}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.5

Add $1$ and $1$ .

$y = - \frac{42 \sqrt{21}}{{\sqrt{21}}^{2}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 12.2.1.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$y = - \frac{42 \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = - \frac{42 \sqrt{21}}{21^{\frac{1}{2} \cdot 2}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = - \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6.4

Cancel the common factor of $2$ .

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Step 12.2.1.1.3.6.4.1

Cancel the common factor.

$y = - \frac{42 \sqrt{21}}{21^{\frac{2}{2}}} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6.4.2

Rewrite the expression.

$y = - \frac{42 \sqrt{21}}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

$y = - \frac{42 \sqrt{21}}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.3.6.5

Evaluate the exponent.

$y = - \frac{42 \sqrt{21}}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

$y = - \frac{42 \sqrt{21}}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

$y = - \frac{42 \sqrt{21}}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.4

Cancel the common factor of $42$ and $21$ .

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Step 12.2.1.1.4.1

Factor $21$ out of $42 \sqrt{21}$ .

$y = - \frac{21 (2 \sqrt{21})}{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.4.2

Cancel the common factors.

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Step 12.2.1.1.4.2.1

Factor $21$ out of $21$ .

$y = - \frac{21 (2 \sqrt{21})}{21 (1)} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.4.2.2

Cancel the common factor.

$y = - \frac{21 (2 \sqrt{21})}{21 \cdot 1} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.4.2.3

Rewrite the expression.

$y = - \frac{2 \sqrt{21}}{1} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.4.2.4

Divide $2 \sqrt{21}$ by $1$ .

$y = - (2 \sqrt{21}) - (- \sqrt{21})$

$x = - \sqrt{21}$

$y = - (2 \sqrt{21}) - (- \sqrt{21})$

$x = - \sqrt{21}$

$y = - (2 \sqrt{21}) - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.5

Multiply $2$ by $- 1$ .

$y = - 2 \sqrt{21} - (- \sqrt{21})$

$x = - \sqrt{21}$

Step 12.2.1.1.6

Multiply $- (- \sqrt{21})$ .

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Step 12.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$y = - 2 \sqrt{21} + 1 \sqrt{21}$

$x = - \sqrt{21}$

Step 12.2.1.1.6.2

Multiply $\sqrt{21}$ by $1$ .

$y = - 2 \sqrt{21} + \sqrt{21}$

$x = - \sqrt{21}$

$y = - 2 \sqrt{21} + \sqrt{21}$

$x = - \sqrt{21}$

$y = - 2 \sqrt{21} + \sqrt{21}$

$x = - \sqrt{21}$

Step 12.2.1.2

Add $- 2 \sqrt{21}$ and $\sqrt{21}$ .

$y = - \sqrt{21}$

$x = - \sqrt{21}$

$y = - \sqrt{21}$

$x = - \sqrt{21}$

$y = - \sqrt{21}$

$x = - \sqrt{21}$

$y = - \sqrt{21}$

$x = - \sqrt{21}$

Step 13

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(6, 1)$

$(- 6, - 1)$

$(\sqrt{21}, \sqrt{21})$

$(- \sqrt{21}, - \sqrt{21})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(6, 1), (- 6, - 1), (\sqrt{21}, \sqrt{21}), (- \sqrt{21}, - \sqrt{21})$

Equation Form:

$x = 6, y = 1$

$x = - 6, y = - 1$

$x = \sqrt{21}, y = \sqrt{21}$

$x = - \sqrt{21}, y = - \sqrt{21}$

Step 15