Finite Math Examples

$p^{2} + 16 q = 1400$ , $900 - p^{2} + 6 q = 0$

Step 1

Solve for $q$ in $p^{2} + 16 q = 1400$ .

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Step 1.1

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

$16 q = 1400 - p^{2}$

$900 - p^{2} + 6 q = 0$

Step 1.2

Divide each term in $16 q = 1400 - p^{2}$ by $16$ and simplify.

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Step 1.2.1

Divide each term in $16 q = 1400 - p^{2}$ by $16$ .

$\frac{16 q}{16} = \frac{1400}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $16$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{16 q}{16} = \frac{1400}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{1400}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{1400}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{1400}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $1400$ and $16$ .

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Step 1.2.3.1.1.1

Factor $8$ out of $1400$ .

$q = \frac{8 (175)}{16} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $8$ out of $16$ .

$q = \frac{8 \cdot 175}{8 \cdot 2} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$q = \frac{8 \cdot 175}{8 \cdot 2} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$q = \frac{175}{2} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} + \frac{- p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 6 q = 0$

Step 2

Replace all occurrences of $q$ with $\frac{175}{2} - \frac{p^{2}}{16}$ in each equation.

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Step 2.1

Replace all occurrences of $q$ in $900 - p^{2} + 6 q = 0$ with $\frac{175}{2} - \frac{p^{2}}{16}$ .

$900 - p^{2} + 6 (\frac{175}{2} - \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $900 - p^{2} + 6 (\frac{175}{2} - \frac{p^{2}}{16})$ .

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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.

$900 - p^{2} + 6 (\frac{175}{2}) + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Factor $2$ out of $6$ .

$900 - p^{2} + 2 (3) (\frac{175}{2}) + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.2.2

Cancel the common factor.

$900 - p^{2} + 2 \cdot (3 (\frac{175}{2})) + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.2.3

Rewrite the expression.

$900 - p^{2} + 3 \cdot 175 + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 3 \cdot 175 + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.3

Multiply $3$ by $175$ .

$900 - p^{2} + 525 + 6 (- \frac{p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{p^{2}}{16}$ into the numerator.

$900 - p^{2} + 525 + 6 (\frac{- p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.4.2

Factor $2$ out of $6$ .

$900 - p^{2} + 525 + 2 (3) (\frac{- p^{2}}{16}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.4.3

Factor $2$ out of $16$ .

$900 - p^{2} + 525 + 2 \cdot (3 (\frac{- p^{2}}{2 \cdot 8})) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.4.4

Cancel the common factor.

$900 - p^{2} + 525 + 2 \cdot (3 (\frac{- p^{2}}{2 \cdot 8})) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.4.5

Rewrite the expression.

$900 - p^{2} + 525 + 3 (\frac{- p^{2}}{8}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 525 + 3 (\frac{- p^{2}}{8}) = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.5

Combine $3$ and $\frac{- p^{2}}{8}$ .

$900 - p^{2} + 525 + \frac{3 (- p^{2})}{8} = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.6

Multiply $- 1$ by $3$ .

$900 - p^{2} + 525 + \frac{- 3 p^{2}}{8} = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.1.7

Move the negative in front of the fraction.

$900 - p^{2} + 525 - \frac{3 p^{2}}{8} = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$900 - p^{2} + 525 - \frac{3 p^{2}}{8} = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.2

Add $900$ and $525$ .

$- p^{2} + 1425 - \frac{3 p^{2}}{8} = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.3

To write $- p^{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- p^{2} \cdot \frac{8}{8} - \frac{3 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.4

Simplify terms.

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Step 2.2.1.4.1

Combine $- p^{2}$ and $\frac{8}{8}$ .

$\frac{- p^{2} \cdot 8}{8} - \frac{3 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.4.2

Combine the numerators over the common denominator.

$\frac{- p^{2} \cdot 8 - 3 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$\frac{- p^{2} \cdot 8 - 3 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5

Simplify each term.

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Step 2.2.1.5.1

Simplify the numerator.

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Step 2.2.1.5.1.1

Factor $p^{2}$ out of $- p^{2} \cdot 8 - 3 p^{2}$ .

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Step 2.2.1.5.1.1.1

Factor $p^{2}$ out of $- p^{2} \cdot 8$ .

$\frac{p^{2} (- 1 \cdot 8) - 3 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.1.1.2

Factor $p^{2}$ out of $- 3 p^{2}$ .

$\frac{p^{2} (- 1 \cdot 8) + p^{2} \cdot - 3}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.1.1.3

Factor $p^{2}$ out of $p^{2} (- 1 \cdot 8) + p^{2} \cdot - 3$ .

$\frac{p^{2} (- 1 \cdot 8 - 3)}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$\frac{p^{2} (- 1 \cdot 8 - 3)}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.1.2

Multiply $- 1$ by $8$ .

$\frac{p^{2} (- 8 - 3)}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.1.3

Subtract $3$ from $- 8$ .

$\frac{p^{2} \cdot - 11}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$\frac{p^{2} \cdot - 11}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.2

Move $- 11$ to the left of $p^{2}$ .

$\frac{- 11 \cdot p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 2.2.1.5.3

Move the negative in front of the fraction.

$- \frac{11 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$- \frac{11 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$- \frac{11 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$- \frac{11 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$- \frac{11 p^{2}}{8} + 1425 = 0$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3

Solve for $p$ in $- \frac{11 p^{2}}{8} + 1425 = 0$ .

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Step 3.1

Subtract $1425$ from both sides of the equation.

$- \frac{11 p^{2}}{8} = - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.2

Multiply both sides of the equation by $- \frac{8}{11}$ .

$- \frac{8}{11} \cdot (- \frac{11 p^{2}}{8}) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{8}{11} (- \frac{11 p^{2}}{8})$ .

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Step 3.3.1.1.1

Cancel the common factor of $8$ .

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Step 3.3.1.1.1.1

Move the leading negative in $- \frac{8}{11}$ into the numerator.

$\frac{- 8}{11} \cdot (- \frac{11 p^{2}}{8}) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{11 p^{2}}{8}$ into the numerator.

$\frac{- 8}{11} \cdot \frac{- 11 p^{2}}{8} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.1.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{11} \cdot \frac{- 11 p^{2}}{8} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{8 \cdot - 1}{11} \cdot \frac{- 11 p^{2}}{8} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{11} \cdot (- 11 p^{2}) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$\frac{- 1}{11} \cdot (- 11 p^{2}) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.2

Cancel the common factor of $11$ .

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Step 3.3.1.1.2.1

Factor $11$ out of $- 11 p^{2}$ .

$\frac{- 1}{11} \cdot (11 (- p^{2})) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{11} \cdot (11 (- p^{2})) = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.2.3

Rewrite the expression.

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.3

Multiply.

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Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.1.1.3.2

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

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = - \frac{8}{11} \cdot - 1425$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- \frac{8}{11} \cdot - 1425$ .

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Step 3.3.2.1.1

Multiply $- 1425$ by $- 1$ .

$p^{2} = 1425 (\frac{8}{11})$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.2.1.2

Combine $1425$ and $\frac{8}{11}$ .

$p^{2} = \frac{1425 \cdot 8}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.3.2.1.3

Multiply $1425$ by $8$ .

$p^{2} = \frac{11400}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = \frac{11400}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = \frac{11400}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p^{2} = \frac{11400}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.4

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

$p = \pm \sqrt{\frac{11400}{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5

Simplify $\pm \sqrt{\frac{11400}{11}}$ .

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Step 3.5.1

Rewrite $\sqrt{\frac{11400}{11}}$ as $\frac{\sqrt{11400}}{\sqrt{11}}$ .

$p = \pm \frac{\sqrt{11400}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.2

Simplify the numerator.

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Step 3.5.2.1

Rewrite $11400$ as $10^{2} \cdot 114$ .

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Step 3.5.2.1.1

Factor $100$ out of $11400$ .

$p = \pm \frac{\sqrt{100 (114)}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.2.1.2

Rewrite $100$ as $10^{2}$ .

$p = \pm \frac{\sqrt{10^{2} \cdot 114}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{\sqrt{10^{2} \cdot 114}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.2.2

Pull terms out from under the radical.

$p = \pm \frac{10 \sqrt{114}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{114}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.3

Multiply $\frac{10 \sqrt{114}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$p = \pm \frac{10 \sqrt{114}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{10 \sqrt{114}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.2

Raise $\sqrt{11}$ to the power of $1$ .

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.3

Raise $\sqrt{11}$ to the power of $1$ .

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{\sqrt{11} \sqrt{11}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.4

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

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.5

Add $1$ and $1$ .

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{{\sqrt{11}}^{2}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 3.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6.2

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

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6.3

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

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11^{\frac{2}{2}}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6.4

Cancel the common factor of $2$ .

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Step 3.5.4.6.4.1

Cancel the common factor.

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11^{\frac{2}{2}}}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6.4.2

Rewrite the expression.

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.4.6.5

Evaluate the exponent.

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{114} \sqrt{11}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.5

Simplify the numerator.

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Step 3.5.5.1

Combine using the product rule for radicals.

$p = \pm \frac{10 \sqrt{11 \cdot 114}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.5.5.2

Multiply $11$ by $114$ .

$p = \pm \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \pm \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.6

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

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Step 3.6.1

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

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.6.2

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

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 3.6.3

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

$p = \frac{10 \sqrt{1254}}{11}, - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \frac{10 \sqrt{1254}}{11}, - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

$p = \frac{10 \sqrt{1254}}{11}, - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{p^{2}}{16}$

Step 4

Replace all occurrences of $p$ with $\frac{10 \sqrt{1254}}{11}$ in each equation.

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Step 4.1

Replace all occurrences of $p$ in $q = \frac{175}{2} - \frac{p^{2}}{16}$ with $\frac{10 \sqrt{1254}}{11}$ .

$q = \frac{175}{2} - \frac{{(\frac{10 \sqrt{1254}}{11})}^{2}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{175}{2} - \frac{{(\frac{10 \sqrt{1254}}{11})}^{2}}{16}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Simplify the numerator.

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Step 4.2.1.1.1.1

Apply the product rule to $\frac{10 \sqrt{1254}}{11}$ .

$q = \frac{175}{2} - \frac{\frac{{(10 \sqrt{1254})}^{2}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2

Simplify the numerator.

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Step 4.2.1.1.1.2.1

Apply the product rule to $10 \sqrt{1254}$ .

$q = \frac{175}{2} - \frac{\frac{10^{2} {\sqrt{1254}}^{2}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.2

Raise $10$ to the power of $2$ .

$q = \frac{175}{2} - \frac{\frac{100 {\sqrt{1254}}^{2}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3

Rewrite ${\sqrt{1254}}^{2}$ as $1254$ .

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Step 4.2.1.1.1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1254}$ as $1254^{\frac{1}{2}}$ .

$q = \frac{175}{2} - \frac{\frac{100 {(1254^{\frac{1}{2}})}^{2}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3.2

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

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254^{\frac{1}{2} \cdot 2}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3.3

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

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254^{\frac{2}{2}}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.2.3.4.1

Cancel the common factor.

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254^{\frac{2}{2}}}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3.4.2

Rewrite the expression.

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.2.3.5

Evaluate the exponent.

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{11^{2}}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.3

Raise $11$ to the power of $2$ .

$q = \frac{175}{2} - \frac{\frac{100 \cdot 1254}{121}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.4

Multiply $100$ by $1254$ .

$q = \frac{175}{2} - \frac{\frac{125400}{121}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.5

Cancel the common factor of $125400$ and $121$ .

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Step 4.2.1.1.1.5.1

Factor $11$ out of $125400$ .

$q = \frac{175}{2} - \frac{\frac{11 (11400)}{121}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.5.2

Cancel the common factors.

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Step 4.2.1.1.1.5.2.1

Factor $11$ out of $121$ .

$q = \frac{175}{2} - \frac{\frac{11 \cdot 11400}{11 \cdot 11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.5.2.2

Cancel the common factor.

$q = \frac{175}{2} - \frac{\frac{11 \cdot 11400}{11 \cdot 11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.1.5.2.3

Rewrite the expression.

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$q = \frac{175}{2} - (\frac{11400}{11} \cdot \frac{1}{16})$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.3

Cancel the common factor of $8$ .

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Step 4.2.1.1.3.1

Factor $8$ out of $11400$ .

$q = \frac{175}{2} - (\frac{8 (1425)}{11} \cdot \frac{1}{16})$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.3.2

Factor $8$ out of $16$ .

$q = \frac{175}{2} - (\frac{8 \cdot 1425}{11} \cdot \frac{1}{8 \cdot 2})$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.3.3

Cancel the common factor.

$q = \frac{175}{2} - (\frac{8 \cdot 1425}{11} \cdot \frac{1}{8 \cdot 2})$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.3.4

Rewrite the expression.

$q = \frac{175}{2} - (\frac{1425}{11} \cdot \frac{1}{2})$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - (\frac{1425}{11} \cdot \frac{1}{2})$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.4

Multiply $\frac{1425}{11}$ by $\frac{1}{2}$ .

$q = \frac{175}{2} - \frac{1425}{11 \cdot 2}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.1.5

Multiply $11$ by $2$ .

$q = \frac{175}{2} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.2

To write $\frac{175}{2}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$q = \frac{175}{2} \cdot \frac{11}{11} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.3

Write each expression with a common denominator of $22$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{175}{2}$ by $\frac{11}{11}$ .

$q = \frac{175 \cdot 11}{2 \cdot 11} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.3.2

Multiply $2$ by $11$ .

$q = \frac{175 \cdot 11}{22} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{175 \cdot 11}{22} - \frac{1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$q = \frac{175 \cdot 11 - 1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $175$ by $11$ .

$q = \frac{1925 - 1425}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.5.2

Subtract $1425$ from $1925$ .

$q = \frac{500}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{500}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.6

Cancel the common factor of $500$ and $22$ .

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Step 4.2.1.6.1

Factor $2$ out of $500$ .

$q = \frac{2 (250)}{22}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.6.2

Cancel the common factors.

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Step 4.2.1.6.2.1

Factor $2$ out of $22$ .

$q = \frac{2 \cdot 250}{2 \cdot 11}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.6.2.2

Cancel the common factor.

$q = \frac{2 \cdot 250}{2 \cdot 11}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 4.2.1.6.2.3

Rewrite the expression.

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = \frac{10 \sqrt{1254}}{11}$

Step 5

Replace all occurrences of $p$ with $- \frac{10 \sqrt{1254}}{11}$ in each equation.

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Step 5.1

Replace all occurrences of $p$ in $q = \frac{175}{2} - \frac{p^{2}}{16}$ with $- \frac{10 \sqrt{1254}}{11}$ .

$q = \frac{175}{2} - \frac{{(- \frac{10 \sqrt{1254}}{11})}^{2}}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{175}{2} - \frac{{(- \frac{10 \sqrt{1254}}{11})}^{2}}{16}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Simplify the numerator.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{10 \sqrt{1254}}{11}$ .

$q = \frac{175}{2} - \frac{{(- 1)}^{2} {(\frac{10 \sqrt{1254}}{11})}^{2}}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.2

Raise $- 1$ to the power of $2$ .

$q = \frac{175}{2} - \frac{1 {(\frac{10 \sqrt{1254}}{11})}^{2}}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.3.1

Apply the product rule to $\frac{10 \sqrt{1254}}{11}$ .

$q = \frac{175}{2} - \frac{1 (\frac{{(10 \sqrt{1254})}^{2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.3.2

Apply the product rule to $10 \sqrt{1254}$ .

$q = \frac{175}{2} - \frac{1 (\frac{10^{2} {\sqrt{1254}}^{2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{10^{2} {\sqrt{1254}}^{2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4

Simplify the numerator.

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Step 5.2.1.1.1.4.1

Raise $10$ to the power of $2$ .

$q = \frac{175}{2} - \frac{1 (\frac{100 {\sqrt{1254}}^{2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2

Rewrite ${\sqrt{1254}}^{2}$ as $1254$ .

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Step 5.2.1.1.1.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1254}$ as $1254^{\frac{1}{2}}$ .

$q = \frac{175}{2} - \frac{1 (\frac{100 {(1254^{\frac{1}{2}})}^{2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2.2

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

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254^{\frac{1}{2} \cdot 2}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2.3

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

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254^{\frac{2}{2}}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2.4

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.4.2.4.1

Cancel the common factor.

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254^{\frac{2}{2}}}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2.4.2

Rewrite the expression.

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.4.2.5

Evaluate the exponent.

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{11^{2}})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.5

Raise $11$ to the power of $2$ .

$q = \frac{175}{2} - \frac{1 (\frac{100 \cdot 1254}{121})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.6

Multiply $100$ by $1254$ .

$q = \frac{175}{2} - \frac{1 (\frac{125400}{121})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.7

Cancel the common factor of $125400$ and $121$ .

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Step 5.2.1.1.1.7.1

Factor $11$ out of $125400$ .

$q = \frac{175}{2} - \frac{1 (\frac{11 (11400)}{121})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.7.2

Cancel the common factors.

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Step 5.2.1.1.1.7.2.1

Factor $11$ out of $121$ .

$q = \frac{175}{2} - \frac{1 (\frac{11 \cdot 11400}{11 \cdot 11})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.7.2.2

Cancel the common factor.

$q = \frac{175}{2} - \frac{1 (\frac{11 \cdot 11400}{11 \cdot 11})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.7.2.3

Rewrite the expression.

$q = \frac{175}{2} - \frac{1 (\frac{11400}{11})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{11400}{11})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1 (\frac{11400}{11})}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.1.8

Multiply $\frac{11400}{11}$ by $1$ .

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{\frac{11400}{11}}{16}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$q = \frac{175}{2} - (\frac{11400}{11} \cdot \frac{1}{16})$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.3

Cancel the common factor of $8$ .

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Step 5.2.1.1.3.1

Factor $8$ out of $11400$ .

$q = \frac{175}{2} - (\frac{8 (1425)}{11} \cdot \frac{1}{16})$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.3.2

Factor $8$ out of $16$ .

$q = \frac{175}{2} - (\frac{8 \cdot 1425}{11} \cdot \frac{1}{8 \cdot 2})$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.3.3

Cancel the common factor.

$q = \frac{175}{2} - (\frac{8 \cdot 1425}{11} \cdot \frac{1}{8 \cdot 2})$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.3.4

Rewrite the expression.

$q = \frac{175}{2} - (\frac{1425}{11} \cdot \frac{1}{2})$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - (\frac{1425}{11} \cdot \frac{1}{2})$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.4

Multiply $\frac{1425}{11}$ by $\frac{1}{2}$ .

$q = \frac{175}{2} - \frac{1425}{11 \cdot 2}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.1.5

Multiply $11$ by $2$ .

$q = \frac{175}{2} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175}{2} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.2

To write $\frac{175}{2}$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$q = \frac{175}{2} \cdot \frac{11}{11} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.3

Write each expression with a common denominator of $22$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{175}{2}$ by $\frac{11}{11}$ .

$q = \frac{175 \cdot 11}{2 \cdot 11} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.3.2

Multiply $2$ by $11$ .

$q = \frac{175 \cdot 11}{22} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{175 \cdot 11}{22} - \frac{1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$q = \frac{175 \cdot 11 - 1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $175$ by $11$ .

$q = \frac{1925 - 1425}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.5.2

Subtract $1425$ from $1925$ .

$q = \frac{500}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{500}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.6

Cancel the common factor of $500$ and $22$ .

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Step 5.2.1.6.1

Factor $2$ out of $500$ .

$q = \frac{2 (250)}{22}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.6.2

Cancel the common factors.

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Step 5.2.1.6.2.1

Factor $2$ out of $22$ .

$q = \frac{2 \cdot 250}{2 \cdot 11}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.6.2.2

Cancel the common factor.

$q = \frac{2 \cdot 250}{2 \cdot 11}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 5.2.1.6.2.3

Rewrite the expression.

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}$

$p = - \frac{10 \sqrt{1254}}{11}$

Step 6

List all of the solutions.

$q = \frac{250}{11}, p = \frac{10 \sqrt{1254}}{11}$

$q = \frac{250}{11}, p = - \frac{10 \sqrt{1254}}{11}$