Calculus Examples

$y = x + 1$ , $y = 9 - x^{2}$ , $x = - 1$ , $x = 2$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x + 1 = 9 - x^{2}$

Step 1.2

Solve $x + 1 = 9 - x^{2}$ for $x$ .

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Step 1.2.1

Add $x^{2}$ to both sides of the equation.

$x + 1 + x^{2} = 9$

Step 1.2.2

Move all terms to the left side of the equation and simplify.

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Step 1.2.2.1

Subtract $9$ from both sides of the equation.

$x + 1 + x^{2} - 9 = 0$

Step 1.2.2.2

Subtract $9$ from $1$ .

$x + x^{2} - 8 = 0$

Step 1.2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 9 - x^{2}$

Step 1.2.4

Substitute the values $a = 1$ , $b = 1$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

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

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.5.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{33}}{2}$

Step 1.2.6.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{33}}{2}$

Step 1.2.6.5

Factor $- 1$ out of $\sqrt{33}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{33})}{2}$

Step 1.2.6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{33})$ .

$x = \frac{- 1 (1 - \sqrt{33})}{2}$

Step 1.2.6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{33}}{2}$

Step 1.2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{33}}{2}$

Step 1.2.7.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{33}}{2}$

Step 1.2.7.5

Factor $- 1$ out of $- \sqrt{33}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{33})}{2}$

Step 1.2.7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{33})$ .

$x = \frac{- 1 (1 + \sqrt{33})}{2}$

Step 1.2.7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{33}}{2}$

Step 1.2.8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{33}}{2}, - \frac{1 + \sqrt{33}}{2}$

Step 1.3

Evaluate $y$ when $x = - \frac{1 - \sqrt{33}}{2}$ .

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Step 1.3.1

Substitute $- \frac{1 - \sqrt{33}}{2}$ for $x$ .

$y = 9 - {(- \frac{1 - \sqrt{33}}{2})}^{2}$

Step 1.3.2

Simplify $9 - {(- \frac{1 - \sqrt{33}}{2})}^{2}$ .

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

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

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Step 1.3.2.1.1.1

Apply the product rule to $- \frac{1 - \sqrt{33}}{2}$ .

$y = 9 - ({(- 1)}^{2} {(\frac{1 - \sqrt{33}}{2})}^{2})$

Step 1.3.2.1.1.2

Apply the product rule to $\frac{1 - \sqrt{33}}{2}$ .

$y = 9 - ({(- 1)}^{2} \frac{{(1 - \sqrt{33})}^{2}}{2^{2}})$

Step 1.3.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.3.2.1.2.1

Move ${(- 1)}^{2}$ .

$y = 9 + {(- 1)}^{2} \cdot - 1 \frac{{(1 - \sqrt{33})}^{2}}{2^{2}}$

Step 1.3.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.3.2.1.2.2.1

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

$y = 9 + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{{(1 - \sqrt{33})}^{2}}{2^{2}}$

Step 1.3.2.1.2.2.2

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

$y = 9 + {(- 1)}^{2 + 1} \frac{{(1 - \sqrt{33})}^{2}}{2^{2}}$

Step 1.3.2.1.2.3

Add $2$ and $1$ .

$y = 9 + {(- 1)}^{3} \frac{{(1 - \sqrt{33})}^{2}}{2^{2}}$

Step 1.3.2.1.3

Raise $- 1$ to the power of $3$ .

$y = 9 - \frac{{(1 - \sqrt{33})}^{2}}{2^{2}}$

Step 1.3.2.1.4

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

$y = 9 - \frac{{(1 - \sqrt{33})}^{2}}{4}$

Step 1.3.2.1.5

Rewrite ${(1 - \sqrt{33})}^{2}$ as $(1 - \sqrt{33}) (1 - \sqrt{33})$ .

$y = 9 - \frac{(1 - \sqrt{33}) (1 - \sqrt{33})}{4}$

Step 1.3.2.1.6

Expand $(1 - \sqrt{33}) (1 - \sqrt{33})$ using the FOIL Method.

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Step 1.3.2.1.6.1

Apply the distributive property.

$y = 9 - \frac{1 (1 - \sqrt{33}) - \sqrt{33} (1 - \sqrt{33})}{4}$

Step 1.3.2.1.6.2

Apply the distributive property.

$y = 9 - \frac{1 \cdot 1 + 1 (- \sqrt{33}) - \sqrt{33} (1 - \sqrt{33})}{4}$

Step 1.3.2.1.6.3

Apply the distributive property.

$y = 9 - \frac{1 \cdot 1 + 1 (- \sqrt{33}) - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4}$

Step 1.3.2.1.7

Simplify and combine like terms.

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Step 1.3.2.1.7.1

Simplify each term.

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Step 1.3.2.1.7.1.1

Multiply $1$ by $1$ .

$y = 9 - \frac{1 + 1 (- \sqrt{33}) - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4}$

Step 1.3.2.1.7.1.2

Multiply $- \sqrt{33}$ by $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4}$

Step 1.3.2.1.7.1.3

Multiply $- 1$ by $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} - \sqrt{33} (- \sqrt{33})}{4}$

Step 1.3.2.1.7.1.4

Multiply $- \sqrt{33} (- \sqrt{33})$ .

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Step 1.3.2.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 1 \sqrt{33} \sqrt{33}}{4}$

Step 1.3.2.1.7.1.4.2

Multiply $\sqrt{33}$ by $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + \sqrt{33} \sqrt{33}}{4}$

Step 1.3.2.1.7.1.4.3

Raise $\sqrt{33}$ to the power of $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1} \sqrt{33}}{4}$

Step 1.3.2.1.7.1.4.4

Raise $\sqrt{33}$ to the power of $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1} {\sqrt{33}}^{1}}{4}$

Step 1.3.2.1.7.1.4.5

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

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1 + 1}}{4}$

Step 1.3.2.1.7.1.4.6

Add $1$ and $1$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{2}}{4}$

Step 1.3.2.1.7.1.5

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 1.3.2.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + {(33^{\frac{1}{2}})}^{2}}{4}$

Step 1.3.2.1.7.1.5.2

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

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 33^{\frac{1}{2} \cdot 2}}{4}$

Step 1.3.2.1.7.1.5.3

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

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 33^{\frac{2}{2}}}{4}$

Step 1.3.2.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 1.3.2.1.7.1.5.4.1

Cancel the common factor.

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 33^{\frac{2}{2}}}{4}$

Step 1.3.2.1.7.1.5.4.2

Rewrite the expression.

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 33^{1}}{4}$

Step 1.3.2.1.7.1.5.5

Evaluate the exponent.

$y = 9 - \frac{1 - \sqrt{33} - \sqrt{33} + 33}{4}$

Step 1.3.2.1.7.2

Add $1$ and $33$ .

$y = 9 - \frac{34 - \sqrt{33} - \sqrt{33}}{4}$

Step 1.3.2.1.7.3

Subtract $\sqrt{33}$ from $- \sqrt{33}$ .

$y = 9 - \frac{34 - 2 \sqrt{33}}{4}$

Step 1.3.2.1.8

Cancel the common factor of $34 - 2 \sqrt{33}$ and $4$ .

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Step 1.3.2.1.8.1

Factor $2$ out of $34$ .

$y = 9 - \frac{2 (17) - 2 \sqrt{33}}{4}$

Step 1.3.2.1.8.2

Factor $2$ out of $- 2 \sqrt{33}$ .

$y = 9 - \frac{2 (17) + 2 (- \sqrt{33})}{4}$

Step 1.3.2.1.8.3

Factor $2$ out of $2 (17) + 2 (- \sqrt{33})$ .

$y = 9 - \frac{2 (17 - \sqrt{33})}{4}$

Step 1.3.2.1.8.4

Cancel the common factors.

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Step 1.3.2.1.8.4.1

Factor $2$ out of $4$ .

$y = 9 - \frac{2 (17 - \sqrt{33})}{2 \cdot 2}$

Step 1.3.2.1.8.4.2

Cancel the common factor.

$y = 9 - \frac{2 (17 - \sqrt{33})}{2 \cdot 2}$

Step 1.3.2.1.8.4.3

Rewrite the expression.

$y = 9 - \frac{17 - \sqrt{33}}{2}$

Step 1.3.2.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 9 \cdot \frac{2}{2} - \frac{17 - \sqrt{33}}{2}$

Step 1.3.2.3

Combine fractions.

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Step 1.3.2.3.1

Combine $9$ and $\frac{2}{2}$ .

$y = \frac{9 \cdot 2}{2} - \frac{17 - \sqrt{33}}{2}$

Step 1.3.2.3.2

Combine the numerators over the common denominator.

$y = \frac{9 \cdot 2 - (17 - \sqrt{33})}{2}$

Step 1.3.2.4

Simplify the numerator.

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Step 1.3.2.4.1

Multiply $9$ by $2$ .

$y = \frac{18 - (17 - \sqrt{33})}{2}$

Step 1.3.2.4.2

Apply the distributive property.

$y = \frac{18 - 1 \cdot 17 - - \sqrt{33}}{2}$

Step 1.3.2.4.3

Multiply $- 1$ by $17$ .

$y = \frac{18 - 17 - - \sqrt{33}}{2}$

Step 1.3.2.4.4

Multiply $- - \sqrt{33}$ .

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Step 1.3.2.4.4.1

Multiply $- 1$ by $- 1$ .

$y = \frac{18 - 17 + 1 \sqrt{33}}{2}$

Step 1.3.2.4.4.2

Multiply $\sqrt{33}$ by $1$ .

$y = \frac{18 - 17 + \sqrt{33}}{2}$

Step 1.3.2.4.5

Subtract $17$ from $18$ .

$y = \frac{1 + \sqrt{33}}{2}$

Step 1.4

Evaluate $y$ when $x = - \frac{1 + \sqrt{33}}{2}$ .

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Step 1.4.1

Substitute $- \frac{1 + \sqrt{33}}{2}$ for $x$ .

$y = 9 - {(- \frac{1 + \sqrt{33}}{2})}^{2}$

Step 1.4.2

Simplify $9 - {(- \frac{1 + \sqrt{33}}{2})}^{2}$ .

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

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

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Step 1.4.2.1.1.1

Apply the product rule to $- \frac{1 + \sqrt{33}}{2}$ .

$y = 9 - ({(- 1)}^{2} {(\frac{1 + \sqrt{33}}{2})}^{2})$

Step 1.4.2.1.1.2

Apply the product rule to $\frac{1 + \sqrt{33}}{2}$ .

$y = 9 - ({(- 1)}^{2} \frac{{(1 + \sqrt{33})}^{2}}{2^{2}})$

Step 1.4.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.4.2.1.2.1

Move ${(- 1)}^{2}$ .

$y = 9 + {(- 1)}^{2} \cdot - 1 \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}$

Step 1.4.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.4.2.1.2.2.1

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

$y = 9 + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}$

Step 1.4.2.1.2.2.2

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

$y = 9 + {(- 1)}^{2 + 1} \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}$

Step 1.4.2.1.2.3

Add $2$ and $1$ .

$y = 9 + {(- 1)}^{3} \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}$

Step 1.4.2.1.3

Raise $- 1$ to the power of $3$ .

$y = 9 - \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}$

Step 1.4.2.1.4

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

$y = 9 - \frac{{(1 + \sqrt{33})}^{2}}{4}$

Step 1.4.2.1.5

Rewrite ${(1 + \sqrt{33})}^{2}$ as $(1 + \sqrt{33}) (1 + \sqrt{33})$ .

$y = 9 - \frac{(1 + \sqrt{33}) (1 + \sqrt{33})}{4}$

Step 1.4.2.1.6

Expand $(1 + \sqrt{33}) (1 + \sqrt{33})$ using the FOIL Method.

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Step 1.4.2.1.6.1

Apply the distributive property.

$y = 9 - \frac{1 (1 + \sqrt{33}) + \sqrt{33} (1 + \sqrt{33})}{4}$

Step 1.4.2.1.6.2

Apply the distributive property.

$y = 9 - \frac{1 \cdot 1 + 1 \sqrt{33} + \sqrt{33} (1 + \sqrt{33})}{4}$

Step 1.4.2.1.6.3

Apply the distributive property.

$y = 9 - \frac{1 \cdot 1 + 1 \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}$

Step 1.4.2.1.7

Simplify and combine like terms.

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Step 1.4.2.1.7.1

Simplify each term.

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Step 1.4.2.1.7.1.1

Multiply $1$ by $1$ .

$y = 9 - \frac{1 + 1 \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}$

Step 1.4.2.1.7.1.2

Multiply $\sqrt{33}$ by $1$ .

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}$

Step 1.4.2.1.7.1.3

Multiply $\sqrt{33}$ by $1$ .

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33} \sqrt{33}}{4}$

Step 1.4.2.1.7.1.4

Combine using the product rule for radicals.

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33 \cdot 33}}{4}$

Step 1.4.2.1.7.1.5

Multiply $33$ by $33$ .

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{1089}}{4}$

Step 1.4.2.1.7.1.6

Rewrite $1089$ as $33^{2}$ .

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33^{2}}}{4}$

Step 1.4.2.1.7.1.7

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

$y = 9 - \frac{1 + \sqrt{33} + \sqrt{33} + 33}{4}$

Step 1.4.2.1.7.2

Add $1$ and $33$ .

$y = 9 - \frac{34 + \sqrt{33} + \sqrt{33}}{4}$

Step 1.4.2.1.7.3

Add $\sqrt{33}$ and $\sqrt{33}$ .

$y = 9 - \frac{34 + 2 \sqrt{33}}{4}$

Step 1.4.2.1.8

Cancel the common factor of $34 + 2 \sqrt{33}$ and $4$ .

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Step 1.4.2.1.8.1

Factor $2$ out of $34$ .

$y = 9 - \frac{2 \cdot 17 + 2 \sqrt{33}}{4}$

Step 1.4.2.1.8.2

Factor $2$ out of $2 \sqrt{33}$ .

$y = 9 - \frac{2 \cdot 17 + 2 (\sqrt{33})}{4}$

Step 1.4.2.1.8.3

Factor $2$ out of $2 \cdot 17 + 2 (\sqrt{33})$ .

$y = 9 - \frac{2 \cdot (17 + \sqrt{33})}{4}$

Step 1.4.2.1.8.4

Cancel the common factors.

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Step 1.4.2.1.8.4.1

Factor $2$ out of $4$ .

$y = 9 - \frac{2 \cdot (17 + \sqrt{33})}{2 (2)}$

Step 1.4.2.1.8.4.2

Cancel the common factor.

$y = 9 - \frac{2 \cdot (17 + \sqrt{33})}{2 \cdot 2}$

Step 1.4.2.1.8.4.3

Rewrite the expression.

$y = 9 - \frac{17 + \sqrt{33}}{2}$

Step 1.4.2.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 9 \cdot \frac{2}{2} - \frac{17 + \sqrt{33}}{2}$

Step 1.4.2.3

Combine fractions.

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Step 1.4.2.3.1

Combine $9$ and $\frac{2}{2}$ .

$y = \frac{9 \cdot 2}{2} - \frac{17 + \sqrt{33}}{2}$

Step 1.4.2.3.2

Combine the numerators over the common denominator.

$y = \frac{9 \cdot 2 - (17 + \sqrt{33})}{2}$

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Multiply $9$ by $2$ .

$y = \frac{18 - (17 + \sqrt{33})}{2}$

Step 1.4.2.4.2

Apply the distributive property.

$y = \frac{18 - 1 \cdot 17 - \sqrt{33}}{2}$

Step 1.4.2.4.3

Multiply $- 1$ by $17$ .

$y = \frac{18 - 17 - \sqrt{33}}{2}$

Step 1.4.2.4.4

Subtract $17$ from $18$ .

$y = \frac{1 - \sqrt{33}}{2}$

Step 1.5

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

$(- \frac{1 - \sqrt{33}}{2}, \frac{1 + \sqrt{33}}{2})$

$(- \frac{1 + \sqrt{33}}{2}, \frac{1 - \sqrt{33}}{2})$

$(- \frac{1 - \sqrt{33}}{2}, \frac{1 + \sqrt{33}}{2})$

$(- \frac{1 + \sqrt{33}}{2}, \frac{1 - \sqrt{33}}{2})$

Step 2

Reorder $9$ and $- x^{2}$ .

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

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 1}^{2} - x^{2} + 9 d x - \int_{- 1}^{2} x + 1 d x$

Step 4

Integrate to find the area between $- 1$ and $2$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- 1}^{2} - x^{2} + 9 - (x + 1) d x$

Step 4.2

Simplify each term.

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Step 4.2.1

Apply the distributive property.

$- x^{2} + 9 - x - 1 \cdot 1$

Step 4.2.2

Multiply $- 1$ by $1$ .

$- x^{2} + 9 - x - 1$

$\int_{- 1}^{2} - x^{2} + 9 - x - 1 d x$

Step 4.3

Subtract $1$ from $9$ .

$\int_{- 1}^{2} - x^{2} - x + 8 d x$

Step 4.4

Split the single integral into multiple integrals.

$\int_{- 1}^{2} - x^{2} d x + \int_{- 1}^{2} - x d x + \int_{- 1}^{2} 8 d x$

Step 4.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{- 1}^{2} x^{2} d x + \int_{- 1}^{2} - x d x + \int_{- 1}^{2} 8 d x$

Step 4.6

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{1}{3} x^{3}]_{- 1}^{2}) + \int_{- 1}^{2} - x d x + \int_{- 1}^{2} 8 d x$

Step 4.7

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{3}}{3}]_{- 1}^{2}) + \int_{- 1}^{2} - x d x + \int_{- 1}^{2} 8 d x$

Step 4.8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- (\frac{x^{3}}{3}]_{- 1}^{2}) - \int_{- 1}^{2} x d x + \int_{- 1}^{2} 8 d x$

Step 4.9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- (\frac{x^{3}}{3}]_{- 1}^{2}) - (\frac{1}{2} x^{2}]_{- 1}^{2}) + \int_{- 1}^{2} 8 d x$

Step 4.10

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

$- (\frac{x^{3}}{3}]_{- 1}^{2}) - (\frac{x^{2}}{2}]_{- 1}^{2}) + \int_{- 1}^{2} 8 d x$

Step 4.11

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- 1}^{2}) - (\frac{x^{2}}{2}]_{- 1}^{2}) + 8 x]_{- 1}^{2}$

Step 4.12

Substitute and simplify.

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Step 4.12.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 1$ .

$- ((\frac{2^{3}}{3}) - \frac{{(- 1)}^{3}}{3}) - (\frac{x^{2}}{2}]_{- 1}^{2}) + 8 x]_{- 1}^{2}$

Step 4.12.2

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $- 1$ .

$- (\frac{2^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + 8 x]_{- 1}^{2}$

Step 4.12.3

Evaluate $8 x$ at $2$ and at $- 1$ .

$- (\frac{2^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4

Simplify.

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Step 4.12.4.1

Raise $2$ to the power of $3$ .

$- (\frac{8}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.2

Raise $- 1$ to the power of $3$ .

$- (\frac{8}{3} - \frac{- 1}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.3

Move the negative in front of the fraction.

$- (\frac{8}{3} - - \frac{1}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.4

Multiply $- 1$ by $- 1$ .

$- (\frac{8}{3} + 1 (\frac{1}{3})) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.5

Multiply $\frac{1}{3}$ by $1$ .

$- (\frac{8}{3} + \frac{1}{3}) - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.6

Combine the numerators over the common denominator.

$- \frac{8 + 1}{3} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.7

Add $8$ and $1$ .

$- \frac{9}{3} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.8

Cancel the common factor of $9$ and $3$ .

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Step 4.12.4.8.1

Factor $3$ out of $9$ .

$- \frac{3 \cdot 3}{3} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.8.2

Cancel the common factors.

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Step 4.12.4.8.2.1

Factor $3$ out of $3$ .

$- \frac{3 \cdot 3}{3 (1)} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.8.2.2

Cancel the common factor.

$- \frac{3 \cdot 3}{3 \cdot 1} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.8.2.3

Rewrite the expression.

$- \frac{3}{1} - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.8.2.4

Divide $3$ by $1$ .

$- 1 \cdot 3 - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.9

Multiply $- 1$ by $3$ .

$- 3 - (\frac{2^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.10

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

$- 3 - (\frac{4}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.11

Cancel the common factor of $4$ and $2$ .

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Step 4.12.4.11.1

Factor $2$ out of $4$ .

$- 3 - (\frac{2 \cdot 2}{2} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.11.2

Cancel the common factors.

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Step 4.12.4.11.2.1

Factor $2$ out of $2$ .

$- 3 - (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.11.2.2

Cancel the common factor.

$- 3 - (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.11.2.3

Rewrite the expression.

$- 3 - (\frac{2}{1} - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.11.2.4

Divide $2$ by $1$ .

$- 3 - (2 - \frac{{(- 1)}^{2}}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.12

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

$- 3 - (2 - \frac{1}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.13

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

$- 3 - (2 \cdot \frac{2}{2} - \frac{1}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.14

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

$- 3 - (\frac{2 \cdot 2}{2} - \frac{1}{2}) + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.15

Combine the numerators over the common denominator.

$- 3 - \frac{2 \cdot 2 - 1}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.16

Simplify the numerator.

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Step 4.12.4.16.1

Multiply $2$ by $2$ .

$- 3 - \frac{4 - 1}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.16.2

Subtract $1$ from $4$ .

$- 3 - \frac{3}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.17

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 \cdot \frac{2}{2} - \frac{3}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.18

Combine $- 3$ and $\frac{2}{2}$ .

$\frac{- 3 \cdot 2}{2} - \frac{3}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.19

Combine the numerators over the common denominator.

$\frac{- 3 \cdot 2 - 3}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.20

Simplify the numerator.

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Step 4.12.4.20.1

Multiply $- 3$ by $2$ .

$\frac{- 6 - 3}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.20.2

Subtract $3$ from $- 6$ .

$\frac{- 9}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.21

Move the negative in front of the fraction.

$- \frac{9}{2} + (8 \cdot 2) - 8 \cdot - 1$

Step 4.12.4.22

Multiply $8$ by $2$ .

$- \frac{9}{2} + 16 - 8 \cdot - 1$

Step 4.12.4.23

Multiply $- 8$ by $- 1$ .

$- \frac{9}{2} + 16 + 8$

Step 4.12.4.24

Add $16$ and $8$ .

$- \frac{9}{2} + 24$

Step 4.12.4.25

To write $24$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{9}{2} + 24 \cdot \frac{2}{2}$

Step 4.12.4.26

Combine $24$ and $\frac{2}{2}$ .

$- \frac{9}{2} + \frac{24 \cdot 2}{2}$

Step 4.12.4.27

Combine the numerators over the common denominator.

$\frac{- 9 + 24 \cdot 2}{2}$

Step 4.12.4.28

Simplify the numerator.

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Step 4.12.4.28.1

Multiply $24$ by $2$ .

$\frac{- 9 + 48}{2}$

Step 4.12.4.28.2

Add $- 9$ and $48$ .

$\frac{39}{2}$

Step 5