Precalculus Examples

$x^{2} - 13 x y + 12 y^{2} = 0$ , $x^{2} + x y = 156$

Step 1

Solve for $y$ in $x^{2} + x y = 156$ .

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Step 1.1

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

$x y = 156 - x^{2}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2

Divide each term in $x y = 156 - x^{2}$ by $x$ and simplify.

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Step 1.2.1

Divide each term in $x y = 156 - x^{2}$ by $x$ .

$\frac{x y}{x} = \frac{156}{x} + \frac{- x^{2}}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

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.

$\frac{x y}{x} = \frac{156}{x} + \frac{- x^{2}}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{156}{x} + \frac{- x^{2}}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} + \frac{- x^{2}}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} + \frac{- x^{2}}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

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 $- x^{2}$ .

$y = \frac{156}{x} + \frac{x (- x)}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

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 $1$ .

$y = \frac{156}{x} + \frac{x (- x)}{x}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2.3.1.2.2

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

$y = \frac{156}{x} + \frac{x (- x)}{x \cdot 1}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2.3.1.2.3

Cancel the common factor.

$y = \frac{156}{x} + \frac{x (- x)}{x \cdot 1}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2.3.1.2.4

Rewrite the expression.

$y = \frac{156}{x} + \frac{- x}{1}$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 1.2.3.1.2.5

Divide $- x$ by $1$ .

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 x y + 12 y^{2} = 0$

Step 2

Replace all occurrences of $y$ with $\frac{156}{x} - x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} - 13 x y + 12 y^{2} = 0$ with $\frac{156}{x} - x$ .

$x^{2} - 13 x (\frac{156}{x} - x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $x^{2} - 13 x (\frac{156}{x} - x) + 12 {(\frac{156}{x} - x)}^{2}$ .

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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.

$x^{2} - 13 x \frac{156}{x} - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

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 $- 13 x$ .

$x^{2} + x \cdot (- 13 \frac{156}{x}) - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.2.2

Cancel the common factor.

$x^{2} + x \cdot (- 13 \frac{156}{x}) - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.2.3

Rewrite the expression.

$x^{2} - 13 \cdot 156 - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 13 \cdot 156 - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.3

Multiply $- 13$ by $156$ .

$x^{2} - 2028 - 13 x (- x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.4

Rewrite using the commutative property of multiplication.

$x^{2} - 2028 - 13 \cdot (- 1 x \cdot x) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

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 $x$ .

$x^{2} - 2028 - 13 \cdot (- 1 (x \cdot x)) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.5.1.2

Multiply $x$ by $x$ .

$x^{2} - 2028 - 13 \cdot (- 1 x^{2}) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 - 13 \cdot (- 1 x^{2}) + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.5.2

Multiply $- 13$ by $- 1$ .

$x^{2} - 2028 + 13 x^{2} + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 {(\frac{156}{x} - x)}^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.6

Rewrite ${(\frac{156}{x} - x)}^{2}$ as $(\frac{156}{x} - x) (\frac{156}{x} - x)$ .

$x^{2} - 2028 + 13 x^{2} + 12 ((\frac{156}{x} - x) (\frac{156}{x} - x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.7

Expand $(\frac{156}{x} - x) (\frac{156}{x} - x)$ using the FOIL Method.

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Step 2.2.1.1.7.1

Apply the distributive property.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{156}{x} \cdot (\frac{156}{x} - x) - x (\frac{156}{x} - x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.7.2

Apply the distributive property.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{156}{x} \cdot \frac{156}{x} + \frac{156}{x} \cdot (- x) - x (\frac{156}{x} - x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.7.3

Apply the distributive property.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{156}{x} \cdot \frac{156}{x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{156}{x} \cdot \frac{156}{x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

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 $\frac{156}{x} \cdot \frac{156}{x}$ .

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Step 2.2.1.1.8.1.1.1

Multiply $\frac{156}{x}$ by $\frac{156}{x}$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{156 \cdot 156}{x \cdot x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.1.2

Multiply $156$ by $156$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x \cdot x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.1.3

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

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x \cdot x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.1.4

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

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x \cdot x} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.1.5

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

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{1 + 1}} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.1.6

Add $1$ and $1$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} + \frac{156}{x} \cdot (- x) - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.2

Rewrite using the commutative property of multiplication.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - \frac{156}{x} \cdot x - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

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 $- \frac{156}{x}$ into the numerator.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} + \frac{- 156}{x} \cdot x - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.3.2

Cancel the common factor.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} + \frac{- 156}{x} \cdot x - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.3.3

Rewrite the expression.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - x \frac{156}{x} - x (- x)) = 0$

$y = \frac{156}{x} - x$

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 $- x$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 + x \cdot (- 1 \frac{156}{x}) - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.4.2

Cancel the common factor.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 + x \cdot (- 1 \frac{156}{x}) - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.4.3

Rewrite the expression.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 1 \cdot 156 - x (- x)) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 1 \cdot 156 - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.5

Multiply $- 1$ by $156$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 - x (- x)) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.6

Rewrite using the commutative property of multiplication.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 - 1 \cdot (- 1 x \cdot x)) = 0$

$y = \frac{156}{x} - x$

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 $x$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 - 1 \cdot (- 1 (x \cdot x))) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.7.2

Multiply $x$ by $x$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 - 1 \cdot (- 1 x^{2})) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 - 1 \cdot (- 1 x^{2})) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.8

Multiply $- 1$ by $- 1$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 + 1 x^{2}) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.1.9

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

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 + x^{2}) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 156 - 156 + x^{2}) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.8.2

Subtract $156$ from $- 156$ .

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 312 + x^{2}) = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}} - 312 + x^{2}) = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.9

Apply the distributive property.

$x^{2} - 2028 + 13 x^{2} + 12 (\frac{24336}{x^{2}}) + 12 \cdot - 312 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.10

Simplify.

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Step 2.2.1.1.10.1

Multiply $12 \frac{24336}{x^{2}}$ .

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Step 2.2.1.1.10.1.1

Combine $12$ and $\frac{24336}{x^{2}}$ .

$x^{2} - 2028 + 13 x^{2} + \frac{12 \cdot 24336}{x^{2}} + 12 \cdot - 312 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.10.1.2

Multiply $12$ by $24336$ .

$x^{2} - 2028 + 13 x^{2} + \frac{292032}{x^{2}} + 12 \cdot - 312 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + \frac{292032}{x^{2}} + 12 \cdot - 312 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.1.10.2

Multiply $12$ by $- 312$ .

$x^{2} - 2028 + 13 x^{2} + \frac{292032}{x^{2}} - 3744 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + \frac{292032}{x^{2}} - 3744 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

$x^{2} - 2028 + 13 x^{2} + \frac{292032}{x^{2}} - 3744 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $x^{2}$ and $13 x^{2}$ .

$14 x^{2} - 2028 + \frac{292032}{x^{2}} - 3744 + 12 x^{2} = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.2.2

Add $14 x^{2}$ and $12 x^{2}$ .

$26 x^{2} - 2028 + \frac{292032}{x^{2}} - 3744 = 0$

$y = \frac{156}{x} - x$

Step 2.2.1.2.3

Subtract $3744$ from $- 2028$ .

$26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$

$y = \frac{156}{x} - x$

$26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$

$y = \frac{156}{x} - x$

$26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$

$y = \frac{156}{x} - x$

$26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$

$y = \frac{156}{x} - x$

$26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$

$y = \frac{156}{x} - x$

Step 3

Solve for $x$ in $26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$ .

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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.

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

$y = \frac{156}{x} - x$

Step 3.1.2

The LCM of one and any expression is the expression.

$x^{2}$

$y = \frac{156}{x} - x$

$x^{2}$

$y = \frac{156}{x} - x$

Step 3.2

Multiply each term in $26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$ by $x^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $26 x^{2} + \frac{292032}{x^{2}} - 5772 = 0$ by $x^{2}$ .

$26 x^{2} x^{2} + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

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 $x^{2}$ .

$26 (x^{2} x^{2}) + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

Step 3.2.2.1.1.2

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

$26 x^{2 + 2} + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

Step 3.2.2.1.1.3

Add $2$ and $2$ .

$26 x^{4} + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

$26 x^{4} + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

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.

$26 x^{4} + \frac{292032}{x^{2}} \cdot x^{2} - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

Step 3.2.2.1.2.2

Rewrite the expression.

$26 x^{4} + 292032 - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

$26 x^{4} + 292032 - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

$26 x^{4} + 292032 - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

$26 x^{4} + 292032 - 5772 x^{2} = 0 x^{2}$

$y = \frac{156}{x} - x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$26 x^{4} + 292032 - 5772 x^{2} = 0$

$y = \frac{156}{x} - x$

$26 x^{4} + 292032 - 5772 x^{2} = 0$

$y = \frac{156}{x} - x$

$26 x^{4} + 292032 - 5772 x^{2} = 0$

$y = \frac{156}{x} - x$

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.

$26 u^{2} - 5772 u + 292032 = 0$

$u = x^{2}$

$y = \frac{156}{x} - x$

Step 3.3.2

Factor the left side of the equation.

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Step 3.3.2.1

Factor $26$ out of $26 u^{2} - 5772 u + 292032$ .

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Step 3.3.2.1.1

Factor $26$ out of $26 u^{2}$ .

$26 (u^{2}) - 5772 u + 292032 = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.1.2

Factor $26$ out of $- 5772 u$ .

$26 (u^{2}) + 26 (- 222 u) + 292032 = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.1.3

Factor $26$ out of $292032$ .

$26 u^{2} + 26 (- 222 u) + 26 \cdot 11232 = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.1.4

Factor $26$ out of $26 u^{2} + 26 (- 222 u)$ .

$26 (u^{2} - 222 u) + 26 \cdot 11232 = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.1.5

Factor $26$ out of $26 (u^{2} - 222 u) + 26 \cdot 11232$ .

$26 (u^{2} - 222 u + 11232) = 0$

$y = \frac{156}{x} - x$

$26 (u^{2} - 222 u + 11232) = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.2

Factor.

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Step 3.3.2.2.1

Factor $u^{2} - 222 u + 11232$ 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 $11232$ and whose sum is $- 222$ .

$- 144, - 78$

$y = \frac{156}{x} - x$

Step 3.3.2.2.1.2

Write the factored form using these integers.

$26 ((u - 144) (u - 78)) = 0$

$y = \frac{156}{x} - x$

$26 ((u - 144) (u - 78)) = 0$

$y = \frac{156}{x} - x$

Step 3.3.2.2.2

Remove unnecessary parentheses.

$26 (u - 144) (u - 78) = 0$

$y = \frac{156}{x} - x$

$26 (u - 144) (u - 78) = 0$

$y = \frac{156}{x} - x$

$26 (u - 144) (u - 78) = 0$

$y = \frac{156}{x} - x$

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 $0$ .

$u - 144 = 0$

$u - 78 = 0$

$y = \frac{156}{x} - x$

Step 3.3.4

Set $u - 144$ equal to $0$ and solve for $u$ .

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Step 3.3.4.1

Set $u - 144$ equal to $0$ .

$u - 144 = 0$

$y = \frac{156}{x} - x$

Step 3.3.4.2

Add $144$ to both sides of the equation.

$u = 144$

$y = \frac{156}{x} - x$

$u = 144$

$y = \frac{156}{x} - x$

Step 3.3.5

Set $u - 78$ equal to $0$ and solve for $u$ .

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Step 3.3.5.1

Set $u - 78$ equal to $0$ .

$u - 78 = 0$

$y = \frac{156}{x} - x$

Step 3.3.5.2

Add $78$ to both sides of the equation.

$u = 78$

$y = \frac{156}{x} - x$

$u = 78$

$y = \frac{156}{x} - x$

Step 3.3.6

The final solution is all the values that make $26 (u - 144) (u - 78) = 0$ true.

$u = 144, 78$

$y = \frac{156}{x} - x$

Step 3.3.7

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

$x^{2} = 144$

$x^{2} = 78$

$y = \frac{156}{x} - x$

Step 3.3.8

Solve the first equation for $x$ .

$x^{2} = 144$

$y = \frac{156}{x} - x$

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.

$x = \pm \sqrt{144}$

$y = \frac{156}{x} - x$

Step 3.3.9.2

Simplify $\pm \sqrt{144}$ .

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Step 3.3.9.2.1

Rewrite $144$ as $12^{2}$ .

$x = \pm \sqrt{12^{2}}$

$y = \frac{156}{x} - x$

Step 3.3.9.2.2

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

$x = \pm 12$

$y = \frac{156}{x} - x$

$x = \pm 12$

$y = \frac{156}{x} - x$

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.

$x = 12$

$y = \frac{156}{x} - x$

Step 3.3.9.3.2

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

$x = - 12$

$y = \frac{156}{x} - x$

Step 3.3.9.3.3

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

$x = 12, - 12$

$y = \frac{156}{x} - x$

$x = 12, - 12$

$y = \frac{156}{x} - x$

$x = 12, - 12$

$y = \frac{156}{x} - x$

Step 3.3.10

Solve the second equation for $x$ .

$x^{2} = 78$

$y = \frac{156}{x} - x$

Step 3.3.11

Solve the equation for $x$ .

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Step 3.3.11.1

Remove parentheses.

$x^{2} = 78$

$y = \frac{156}{x} - x$

Step 3.3.11.2

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

$x = \pm \sqrt{78}$

$y = \frac{156}{x} - x$

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.

$x = \sqrt{78}$

$y = \frac{156}{x} - x$

Step 3.3.11.3.2

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

$x = - \sqrt{78}$

$y = \frac{156}{x} - x$

Step 3.3.11.3.3

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

$x = \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

$x = \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

$x = \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

Step 3.3.12

The solution to $26 x^{4} - 5772 x^{2} + 292032 = 0$ is $x = 12, - 12, \sqrt{78}, - \sqrt{78}$ .

$x = 12, - 12, \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

$x = 12, - 12, \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

$x = 12, - 12, \sqrt{78}, - \sqrt{78}$

$y = \frac{156}{x} - x$

Step 4

Replace all occurrences of $x$ with $12$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $12$ .

$y = \frac{156}{12} - (12)$

$x = 12$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{156}{12} - (12)$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Divide $156$ by $12$ .

$y = 13 - (12)$

$x = 12$

Step 4.2.1.1.2

Multiply $- 1$ by $12$ .

$y = 13 - 12$

$x = 12$

$y = 13 - 12$

$x = 12$

Step 4.2.1.2

Subtract $12$ from $13$ .

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

Step 5

Replace all occurrences of $x$ with $- 12$ in each equation.

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Step 5.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $- 12$ .

$y = \frac{156}{- 12} - (- 12)$

$x = - 12$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{156}{- 12} - (- 12)$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Divide $156$ by $- 12$ .

$y = - 13 - (- 12)$

$x = - 12$

Step 5.2.1.1.2

Multiply $- 1$ by $- 12$ .

$y = - 13 + 12$

$x = - 12$

$y = - 13 + 12$

$x = - 12$

Step 5.2.1.2

Add $- 13$ and $12$ .

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

Step 6

Replace all occurrences of $x$ with $12$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $12$ .

$y = \frac{156}{12} - (12)$

$x = 12$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{156}{12} - (12)$ .

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Step 6.2.1.1

Simplify each term.

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Step 6.2.1.1.1

Divide $156$ by $12$ .

$y = 13 - (12)$

$x = 12$

Step 6.2.1.1.2

Multiply $- 1$ by $12$ .

$y = 13 - 12$

$x = 12$

$y = 13 - 12$

$x = 12$

Step 6.2.1.2

Subtract $12$ from $13$ .

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

Step 7

Replace all occurrences of $x$ with $- 12$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $- 12$ .

$y = \frac{156}{- 12} - (- 12)$

$x = - 12$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{156}{- 12} - (- 12)$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Divide $156$ by $- 12$ .

$y = - 13 - (- 12)$

$x = - 12$

Step 7.2.1.1.2

Multiply $- 1$ by $- 12$ .

$y = - 13 + 12$

$x = - 12$

$y = - 13 + 12$

$x = - 12$

Step 7.2.1.2

Add $- 13$ and $12$ .

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

Step 8

Replace all occurrences of $x$ with $\sqrt{78}$ in each equation.

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Step 8.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $\sqrt{78}$ .

$y = \frac{156}{\sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{156}{\sqrt{78}} - (\sqrt{78})$ .

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Step 8.2.1.1

Simplify each term.

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Step 8.2.1.1.1

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = \frac{156}{\sqrt{78}} \cdot \frac{\sqrt{78}}{\sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2

Combine and simplify the denominator.

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Step 8.2.1.1.2.1

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.2

Raise $\sqrt{78}$ to the power of $1$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.3

Raise $\sqrt{78}$ to the power of $1$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.4

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

$y = \frac{156 \sqrt{78}}{{\sqrt{78}}^{1 + 1}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.5

Add $1$ and $1$ .

$y = \frac{156 \sqrt{78}}{{\sqrt{78}}^{2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.6

Rewrite ${\sqrt{78}}^{2}$ as $78$ .

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Step 8.2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{78}$ as $78^{\frac{1}{2}}$ .

$y = \frac{156 \sqrt{78}}{{(78^{\frac{1}{2}})}^{2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.6.2

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

$y = \frac{156 \sqrt{78}}{78^{\frac{1}{2} \cdot 2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.6.3

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

$y = \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (\sqrt{78})$

$x = \sqrt{78}$

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.

$y = \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.6.4.2

Rewrite the expression.

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.2.6.5

Evaluate the exponent.

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.3

Cancel the common factor of $156$ and $78$ .

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Step 8.2.1.1.3.1

Factor $78$ out of $156 \sqrt{78}$ .

$y = \frac{78 (2 \sqrt{78})}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.3.2

Cancel the common factors.

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Step 8.2.1.1.3.2.1

Factor $78$ out of $78$ .

$y = \frac{78 (2 \sqrt{78})}{78 (1)} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.3.2.2

Cancel the common factor.

$y = \frac{78 (2 \sqrt{78})}{78 \cdot 1} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.3.2.3

Rewrite the expression.

$y = \frac{2 \sqrt{78}}{1} - (\sqrt{78})$

$x = \sqrt{78}$

Step 8.2.1.1.3.2.4

Divide $2 \sqrt{78}$ by $1$ .

$y = 2 \sqrt{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - \sqrt{78}$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - \sqrt{78}$

$x = \sqrt{78}$

Step 8.2.1.2

Subtract $\sqrt{78}$ from $2 \sqrt{78}$ .

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

Step 9

Replace all occurrences of $x$ with $12$ in each equation.

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Step 9.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $12$ .

$y = \frac{156}{12} - (12)$

$x = 12$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\frac{156}{12} - (12)$ .

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Step 9.2.1.1

Simplify each term.

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Step 9.2.1.1.1

Divide $156$ by $12$ .

$y = 13 - (12)$

$x = 12$

Step 9.2.1.1.2

Multiply $- 1$ by $12$ .

$y = 13 - 12$

$x = 12$

$y = 13 - 12$

$x = 12$

Step 9.2.1.2

Subtract $12$ from $13$ .

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

$y = 1$

$x = 12$

Step 10

Replace all occurrences of $x$ with $- 12$ in each equation.

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Step 10.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $- 12$ .

$y = \frac{156}{- 12} - (- 12)$

$x = - 12$

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $\frac{156}{- 12} - (- 12)$ .

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Step 10.2.1.1

Simplify each term.

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Step 10.2.1.1.1

Divide $156$ by $- 12$ .

$y = - 13 - (- 12)$

$x = - 12$

Step 10.2.1.1.2

Multiply $- 1$ by $- 12$ .

$y = - 13 + 12$

$x = - 12$

$y = - 13 + 12$

$x = - 12$

Step 10.2.1.2

Add $- 13$ and $12$ .

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

$y = - 1$

$x = - 12$

Step 11

Replace all occurrences of $x$ with $\sqrt{78}$ in each equation.

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Step 11.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $\sqrt{78}$ .

$y = \frac{156}{\sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2

Simplify the right side.

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Step 11.2.1

Simplify $\frac{156}{\sqrt{78}} - (\sqrt{78})$ .

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Step 11.2.1.1

Simplify each term.

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Step 11.2.1.1.1

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = \frac{156}{\sqrt{78}} \cdot \frac{\sqrt{78}}{\sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2

Combine and simplify the denominator.

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Step 11.2.1.1.2.1

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.2

Raise $\sqrt{78}$ to the power of $1$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.3

Raise $\sqrt{78}$ to the power of $1$ .

$y = \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.4

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

$y = \frac{156 \sqrt{78}}{{\sqrt{78}}^{1 + 1}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.5

Add $1$ and $1$ .

$y = \frac{156 \sqrt{78}}{{\sqrt{78}}^{2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.6

Rewrite ${\sqrt{78}}^{2}$ as $78$ .

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Step 11.2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{78}$ as $78^{\frac{1}{2}}$ .

$y = \frac{156 \sqrt{78}}{{(78^{\frac{1}{2}})}^{2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.6.2

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

$y = \frac{156 \sqrt{78}}{78^{\frac{1}{2} \cdot 2}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.6.3

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

$y = \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (\sqrt{78})$

$x = \sqrt{78}$

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.

$y = \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.6.4.2

Rewrite the expression.

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.2.6.5

Evaluate the exponent.

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = \frac{156 \sqrt{78}}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.3

Cancel the common factor of $156$ and $78$ .

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Step 11.2.1.1.3.1

Factor $78$ out of $156 \sqrt{78}$ .

$y = \frac{78 (2 \sqrt{78})}{78} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.3.2

Cancel the common factors.

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Step 11.2.1.1.3.2.1

Factor $78$ out of $78$ .

$y = \frac{78 (2 \sqrt{78})}{78 (1)} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.3.2.2

Cancel the common factor.

$y = \frac{78 (2 \sqrt{78})}{78 \cdot 1} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.3.2.3

Rewrite the expression.

$y = \frac{2 \sqrt{78}}{1} - (\sqrt{78})$

$x = \sqrt{78}$

Step 11.2.1.1.3.2.4

Divide $2 \sqrt{78}$ by $1$ .

$y = 2 \sqrt{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - (\sqrt{78})$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - \sqrt{78}$

$x = \sqrt{78}$

$y = 2 \sqrt{78} - \sqrt{78}$

$x = \sqrt{78}$

Step 11.2.1.2

Subtract $\sqrt{78}$ from $2 \sqrt{78}$ .

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

$y = \sqrt{78}$

$x = \sqrt{78}$

Step 12

Replace all occurrences of $x$ with $- \sqrt{78}$ in each equation.

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Step 12.1

Replace all occurrences of $x$ in $y = \frac{156}{x} - x$ with $- \sqrt{78}$ .

$y = \frac{156}{- \sqrt{78}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Simplify $\frac{156}{- \sqrt{78}} - (- \sqrt{78})$ .

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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.

$y = - \frac{156}{\sqrt{78}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.2

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = - (\frac{156}{\sqrt{78}} \cdot \frac{\sqrt{78}}{\sqrt{78}}) - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3

Combine and simplify the denominator.

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Step 12.2.1.1.3.1

Multiply $\frac{156}{\sqrt{78}}$ by $\frac{\sqrt{78}}{\sqrt{78}}$ .

$y = - \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.2

Raise $\sqrt{78}$ to the power of $1$ .

$y = - \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.3

Raise $\sqrt{78}$ to the power of $1$ .

$y = - \frac{156 \sqrt{78}}{\sqrt{78} \sqrt{78}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.4

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

$y = - \frac{156 \sqrt{78}}{{\sqrt{78}}^{1 + 1}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.5

Add $1$ and $1$ .

$y = - \frac{156 \sqrt{78}}{{\sqrt{78}}^{2}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.6

Rewrite ${\sqrt{78}}^{2}$ as $78$ .

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Step 12.2.1.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{78}$ as $78^{\frac{1}{2}}$ .

$y = - \frac{156 \sqrt{78}}{{(78^{\frac{1}{2}})}^{2}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.6.2

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

$y = - \frac{156 \sqrt{78}}{78^{\frac{1}{2} \cdot 2}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.6.3

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

$y = - \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (- \sqrt{78})$

$x = - \sqrt{78}$

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.

$y = - \frac{156 \sqrt{78}}{78^{\frac{2}{2}}} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.6.4.2

Rewrite the expression.

$y = - \frac{156 \sqrt{78}}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

$y = - \frac{156 \sqrt{78}}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.3.6.5

Evaluate the exponent.

$y = - \frac{156 \sqrt{78}}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

$y = - \frac{156 \sqrt{78}}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

$y = - \frac{156 \sqrt{78}}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.4

Cancel the common factor of $156$ and $78$ .

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Step 12.2.1.1.4.1

Factor $78$ out of $156 \sqrt{78}$ .

$y = - \frac{78 (2 \sqrt{78})}{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.4.2

Cancel the common factors.

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Step 12.2.1.1.4.2.1

Factor $78$ out of $78$ .

$y = - \frac{78 (2 \sqrt{78})}{78 (1)} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.4.2.2

Cancel the common factor.

$y = - \frac{78 (2 \sqrt{78})}{78 \cdot 1} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.4.2.3

Rewrite the expression.

$y = - \frac{2 \sqrt{78}}{1} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.4.2.4

Divide $2 \sqrt{78}$ by $1$ .

$y = - (2 \sqrt{78}) - (- \sqrt{78})$

$x = - \sqrt{78}$

$y = - (2 \sqrt{78}) - (- \sqrt{78})$

$x = - \sqrt{78}$

$y = - (2 \sqrt{78}) - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.5

Multiply $2$ by $- 1$ .

$y = - 2 \sqrt{78} - (- \sqrt{78})$

$x = - \sqrt{78}$

Step 12.2.1.1.6

Multiply $- (- \sqrt{78})$ .

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Step 12.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$y = - 2 \sqrt{78} + 1 \sqrt{78}$

$x = - \sqrt{78}$

Step 12.2.1.1.6.2

Multiply $\sqrt{78}$ by $1$ .

$y = - 2 \sqrt{78} + \sqrt{78}$

$x = - \sqrt{78}$

$y = - 2 \sqrt{78} + \sqrt{78}$

$x = - \sqrt{78}$

$y = - 2 \sqrt{78} + \sqrt{78}$

$x = - \sqrt{78}$

Step 12.2.1.2

Add $- 2 \sqrt{78}$ and $\sqrt{78}$ .

$y = - \sqrt{78}$

$x = - \sqrt{78}$

$y = - \sqrt{78}$

$x = - \sqrt{78}$

$y = - \sqrt{78}$

$x = - \sqrt{78}$

$y = - \sqrt{78}$

$x = - \sqrt{78}$

Step 13

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

$(12, 1)$

$(- 12, - 1)$

$(\sqrt{78}, \sqrt{78})$

$(- \sqrt{78}, - \sqrt{78})$

Step 14

The result can be shown in multiple forms.

Point Form:

$(12, 1), (- 12, - 1), (\sqrt{78}, \sqrt{78}), (- \sqrt{78}, - \sqrt{78})$

Equation Form:

$x = 12, y = 1$

$x = - 12, y = - 1$

$x = \sqrt{78}, y = \sqrt{78}$

$x = - \sqrt{78}, y = - \sqrt{78}$

Step 15