Calculus Examples

$y = \sqrt{4 x}$ , $y = \frac{x^{2}}{4}$

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.

$\sqrt{4 x} = \frac{x^{2}}{4}$

Step 1.2

Solve $\sqrt{4 x} = \frac{x^{2}}{4}$ for $x$ .

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Step 1.2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 x}}^{2} = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2

Simplify each side of the equation.

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Step 1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x}$ as ${(4 x)}^{\frac{1}{2}}$ .

${({(4 x)}^{\frac{1}{2}})}^{2} = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Simplify ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.2.1.1

Multiply the exponents in ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.2.2.1.1.1

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

${(4 x)}^{\frac{1}{2} \cdot 2} = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1.2.1

Cancel the common factor.

${(4 x)}^{\frac{1}{2} \cdot 2} = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

${(4 x)}^{1} = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2.2.1.2

Simplify.

$4 x = {(\frac{x^{2}}{4})}^{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Simplify ${(\frac{x^{2}}{4})}^{2}$ .

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Step 1.2.2.3.1.1

Apply the product rule to $\frac{x^{2}}{4}$ .

$4 x = \frac{{(x^{2})}^{2}}{4^{2}}$

Step 1.2.2.3.1.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.2.2.3.1.2.1

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

$4 x = \frac{x^{2 \cdot 2}}{4^{2}}$

Step 1.2.2.3.1.2.2

Multiply $2$ by $2$ .

$4 x = \frac{x^{4}}{4^{2}}$

Step 1.2.2.3.1.3

Raise $4$ to the power of $2$ .

$4 x = \frac{x^{4}}{16}$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Multiply both sides by $16$ .

$4 x \cdot 16 = \frac{x^{4}}{16} \cdot 16$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Simplify the left side.

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Step 1.2.3.2.1.1

Multiply $16$ by $4$ .

$64 x = \frac{x^{4}}{16} \cdot 16$

Step 1.2.3.2.2

Simplify the right side.

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Step 1.2.3.2.2.1

Cancel the common factor of $16$ .

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Step 1.2.3.2.2.1.1

Cancel the common factor.

$64 x = \frac{x^{4}}{16} \cdot 16$

Step 1.2.3.2.2.1.2

Rewrite the expression.

$64 x = x^{4}$

Step 1.2.3.3

Solve for $x$ .

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Step 1.2.3.3.1

Subtract $x^{4}$ from both sides of the equation.

$64 x - x^{4} = 0$

Step 1.2.3.3.2

Factor the left side of the equation.

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Step 1.2.3.3.2.1

Factor $x$ out of $64 x - x^{4}$ .

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Step 1.2.3.3.2.1.1

Factor $x$ out of $64 x$ .

$x \cdot 64 - x^{4} = 0$

Step 1.2.3.3.2.1.2

Factor $x$ out of $- x^{4}$ .

$x \cdot 64 + x (- x^{3}) = 0$

Step 1.2.3.3.2.1.3

Factor $x$ out of $x \cdot 64 + x (- x^{3})$ .

$x (64 - x^{3}) = 0$

Step 1.2.3.3.2.2

Rewrite $64$ as $4^{3}$ .

$x (4^{3} - x^{3}) = 0$

Step 1.2.3.3.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 4$ and $b = x$ .

$x ((4 - x) (4^{2} + 4 x + x^{2})) = 0$

Step 1.2.3.3.2.4

Factor.

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Step 1.2.3.3.2.4.1

Raise $4$ to the power of $2$ .

$x ((4 - x) (16 + 4 x + x^{2})) = 0$

Step 1.2.3.3.2.4.2

Remove unnecessary parentheses.

$x (4 - x) (16 + 4 x + x^{2}) = 0$

Step 1.2.3.3.3

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

$x = 0$

$4 - x = 0$

$16 + 4 x + x^{2} = 0 + y = \frac{x^{2}}{4}$

Step 1.2.3.3.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3.5

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.5.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 1.2.3.3.5.2

Solve $4 - x = 0$ for $x$ .

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Step 1.2.3.3.5.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 1.2.3.3.5.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 1.2.3.3.5.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 1.2.3.3.5.2.2.2

Simplify the left side.

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Step 1.2.3.3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 1.2.3.3.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 1.2.3.3.5.2.2.3

Simplify the right side.

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Step 1.2.3.3.5.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 1.2.3.3.6

Set $16 + 4 x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.6.1

Set $16 + 4 x + x^{2}$ equal to $0$ .

$16 + 4 x + x^{2} = 0$

Step 1.2.3.3.6.2

Solve $16 + 4 x + x^{2} = 0$ for $x$ .

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Step 1.2.3.3.6.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.3.3.6.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1} + y = \frac{x^{2}}{4}$

Step 1.2.3.3.6.2.3

Simplify.

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Step 1.2.3.3.6.2.3.1

Simplify the numerator.

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Step 1.2.3.3.6.2.3.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 1.2.3.3.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.2.3.3.6.2.3.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 1.2.3.3.6.2.3.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 1.2.3.3.6.2.4

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

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Step 1.2.3.3.6.2.4.1

Simplify the numerator.

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Step 1.2.3.3.6.2.4.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 1.2.3.3.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.2.3.3.6.2.4.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 1.2.3.3.6.2.4.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 1.2.3.3.6.2.4.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 i \sqrt{3}$

Step 1.2.3.3.6.2.5

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

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Step 1.2.3.3.6.2.5.1

Simplify the numerator.

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Step 1.2.3.3.6.2.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 1.2.3.3.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.3

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.2.3.3.6.2.5.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.3.3.6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2}$

Step 1.2.3.3.6.2.5.3

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{2}$ .

$x = - 2 \pm 2 i \sqrt{3}$

Step 1.2.3.3.6.2.5.4

Change the $\pm$ to $-$ .

$x = - 2 - 2 i \sqrt{3}$

Step 1.2.3.3.6.2.6

The final answer is the combination of both solutions.

$x = - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 1.2.3.3.7

The final solution is all the values that make $x (4 - x) (16 + 4 x + x^{2}) = 0$ true.

$x = 0, 4, - 2 + 2 i \sqrt{3}, - 2 - 2 i \sqrt{3}$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = \frac{{(0)}^{2}}{4}$

Step 1.3.2

Substitute $0$ for $x$ in $y = \frac{{(0)}^{2}}{4}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \frac{0^{2}}{4}$

Step 1.3.2.2

Simplify $\frac{0^{2}}{4}$ .

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Step 1.3.2.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{4}$

Step 1.3.2.2.2

Divide $0$ by $4$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 4$ .

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Step 1.4.1

Substitute $4$ for $x$ .

$y = \frac{{(4)}^{2}}{4}$

Step 1.4.2

Substitute $4$ for $x$ in $y = \frac{{(4)}^{2}}{4}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = \frac{4^{2}}{4}$

Step 1.4.2.2

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.4.2.2.1

Factor $4$ out of $4^{2}$ .

$y = \frac{4 \cdot 4}{4}$

Step 1.4.2.2.2

Cancel the common factors.

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Step 1.4.2.2.2.1

Factor $4$ out of $4$ .

$y = \frac{4 \cdot 4}{4 (1)}$

Step 1.4.2.2.2.2

Cancel the common factor.

$y = \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.4.2.2.2.3

Rewrite the expression.

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

Step 1.4.2.2.2.4

Divide $4$ by $1$ .

$y = 4$

Step 1.5

Evaluate $y$ when $x = - 2 + 2 i \sqrt{3}$ .

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Step 1.5.1

Substitute $- 2 + 2 i \sqrt{3}$ for $x$ .

$y = \frac{{(- 2 + 2 i \sqrt{3})}^{2}}{4}$

Step 1.5.2

Simplify $\frac{{(- 2 + 2 i \sqrt{3})}^{2}}{4}$ .

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Step 1.5.2.1

Rewrite ${(- 2 + 2 i \sqrt{3})}^{2}$ as $(- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})$ .

$y = \frac{(- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})}{4}$

Step 1.5.2.2

Expand $(- 2 + 2 i \sqrt{3}) (- 2 + 2 i \sqrt{3})$ using the FOIL Method.

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Step 1.5.2.2.1

Apply the distributive property.

$y = \frac{- 2 (- 2 + 2 i \sqrt{3}) + 2 i \sqrt{3} (- 2 + 2 i \sqrt{3})}{4}$

Step 1.5.2.2.2

Apply the distributive property.

$y = \frac{- 2 \cdot - 2 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} (- 2 + 2 i \sqrt{3})}{4}$

Step 1.5.2.2.3

Apply the distributive property.

$y = \frac{- 2 \cdot - 2 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})}{4}$

Step 1.5.2.3

Simplify and combine like terms.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Multiply $- 2$ by $- 2$ .

$y = \frac{4 - 2 (2 i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})}{4}$

Step 1.5.2.3.1.2

Multiply $2$ by $- 2$ .

$y = \frac{4 - 4 (i \sqrt{3}) + 2 i \sqrt{3} \cdot - 2 + 2 i \sqrt{3} (2 i \sqrt{3})}{4}$

Step 1.5.2.3.1.3

Multiply $- 2$ by $2$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 2 i \sqrt{3} (2 i \sqrt{3})}{4}$

Step 1.5.2.3.1.4

Multiply $2 i \sqrt{3} (2 i \sqrt{3})$ .

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Step 1.5.2.3.1.4.1

Multiply $2$ by $2$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i \sqrt{3} (i \sqrt{3})}{4}$

Step 1.5.2.3.1.4.2

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 1.5.2.3.1.4.3

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 1.5.2.3.1.4.4

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 1.5.2.3.1.4.5

Add $1$ and $1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 1.5.2.3.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 1.5.2.3.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 1.5.2.3.1.4.8

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 1.5.2.3.1.4.9

Add $1$ and $1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{2}}{4}$

Step 1.5.2.3.1.5

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} + 4 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 1.5.2.3.1.6

Multiply $4$ by $- 1$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 {\sqrt{3}}^{2}}{4}$

Step 1.5.2.3.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.5.2.3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 1.5.2.3.1.7.2

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 1.5.2.3.1.7.3

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

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}{4}$

Step 1.5.2.3.1.7.4

Cancel the common factor of $2$ .

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Step 1.5.2.3.1.7.4.1

Cancel the common factor.

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}{4}$

Step 1.5.2.3.1.7.4.2

Rewrite the expression.

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3^{1}}{4}$

Step 1.5.2.3.1.7.5

Evaluate the exponent.

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 4 \cdot 3}{4}$

Step 1.5.2.3.1.8

Multiply $- 4$ by $3$ .

$y = \frac{4 - 4 i \sqrt{3} - 4 i \sqrt{3} - 12}{4}$

Step 1.5.2.3.2

Subtract $12$ from $4$ .

$y = \frac{- 4 i \sqrt{3} - 4 i \sqrt{3} - 8}{4}$

Step 1.5.2.3.3

Subtract $4 i \sqrt{3}$ from $- 4 i \sqrt{3}$ .

$y = \frac{- 8 i \sqrt{3} - 8}{4}$

Step 1.5.2.4

Reorder $- 8 i \sqrt{3}$ and $- 8$ .

$y = \frac{- 8 - 8 i \sqrt{3}}{4}$

Step 1.5.2.5

Cancel the common factor of $- 8 - 8 i \sqrt{3}$ and $4$ .

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Step 1.5.2.5.1

Factor $4$ out of $- 8$ .

$y = \frac{4 \cdot - 2 - 8 i \sqrt{3}}{4}$

Step 1.5.2.5.2

Factor $4$ out of $- 8 i \sqrt{3}$ .

$y = \frac{4 \cdot - 2 + 4 (- 2 i \sqrt{3})}{4}$

Step 1.5.2.5.3

Factor $4$ out of $4 (- 2) + 4 (- 2 i \sqrt{3})$ .

$y = \frac{4 (- 2 - 2 i \sqrt{3})}{4}$

Step 1.5.2.5.4

Cancel the common factors.

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Step 1.5.2.5.4.1

Factor $4$ out of $4$ .

$y = \frac{4 (- 2 - 2 i \sqrt{3})}{4 (1)}$

Step 1.5.2.5.4.2

Cancel the common factor.

$y = \frac{4 (- 2 - 2 i \sqrt{3})}{4 \cdot 1}$

Step 1.5.2.5.4.3

Rewrite the expression.

$y = \frac{- 2 - 2 i \sqrt{3}}{1}$

Step 1.5.2.5.4.4

Divide $- 2 - 2 i \sqrt{3}$ by $1$ .

$y = - 2 - 2 i \sqrt{3}$

Step 1.6

Evaluate $y$ when $x = - 2 - 2 i \sqrt{3}$ .

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Step 1.6.1

Substitute $- 2 - 2 i \sqrt{3}$ for $x$ .

$y = \frac{{(- 2 - 2 i \sqrt{3})}^{2}}{4}$

Step 1.6.2

Simplify $\frac{{(- 2 - 2 i \sqrt{3})}^{2}}{4}$ .

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Step 1.6.2.1

Rewrite ${(- 2 - 2 i \sqrt{3})}^{2}$ as $(- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})$ .

$y = \frac{(- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})}{4}$

Step 1.6.2.2

Expand $(- 2 - 2 i \sqrt{3}) (- 2 - 2 i \sqrt{3})$ using the FOIL Method.

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Step 1.6.2.2.1

Apply the distributive property.

$y = \frac{- 2 (- 2 - 2 i \sqrt{3}) - 2 i \sqrt{3} (- 2 - 2 i \sqrt{3})}{4}$

Step 1.6.2.2.2

Apply the distributive property.

$y = \frac{- 2 \cdot - 2 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} (- 2 - 2 i \sqrt{3})}{4}$

Step 1.6.2.2.3

Apply the distributive property.

$y = \frac{- 2 \cdot - 2 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})}{4}$

Step 1.6.2.3

Simplify and combine like terms.

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Step 1.6.2.3.1

Simplify each term.

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Step 1.6.2.3.1.1

Multiply $- 2$ by $- 2$ .

$y = \frac{4 - 2 (- 2 i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})}{4}$

Step 1.6.2.3.1.2

Multiply $- 2$ by $- 2$ .

$y = \frac{4 + 4 (i \sqrt{3}) - 2 i \sqrt{3} \cdot - 2 - 2 i \sqrt{3} (- 2 i \sqrt{3})}{4}$

Step 1.6.2.3.1.3

Multiply $- 2$ by $- 2$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 2 i \sqrt{3} (- 2 i \sqrt{3})}{4}$

Step 1.6.2.3.1.4

Multiply $- 2 i \sqrt{3} (- 2 i \sqrt{3})$ .

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Step 1.6.2.3.1.4.1

Multiply $- 2$ by $- 2$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i \sqrt{3} (i \sqrt{3})}{4}$

Step 1.6.2.3.1.4.2

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 1.6.2.3.1.4.3

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 1.6.2.3.1.4.4

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 1.6.2.3.1.4.5

Add $1$ and $1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 1.6.2.3.1.4.6

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 1.6.2.3.1.4.7

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 1.6.2.3.1.4.8

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 1.6.2.3.1.4.9

Add $1$ and $1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 i^{2} {\sqrt{3}}^{2}}{4}$

Step 1.6.2.3.1.5

Rewrite $i^{2}$ as $- 1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} + 4 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 1.6.2.3.1.6

Multiply $4$ by $- 1$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 {\sqrt{3}}^{2}}{4}$

Step 1.6.2.3.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.6.2.3.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 1.6.2.3.1.7.2

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 1.6.2.3.1.7.3

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

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}{4}$

Step 1.6.2.3.1.7.4

Cancel the common factor of $2$ .

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Step 1.6.2.3.1.7.4.1

Cancel the common factor.

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}{4}$

Step 1.6.2.3.1.7.4.2

Rewrite the expression.

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3^{1}}{4}$

Step 1.6.2.3.1.7.5

Evaluate the exponent.

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 4 \cdot 3}{4}$

Step 1.6.2.3.1.8

Multiply $- 4$ by $3$ .

$y = \frac{4 + 4 i \sqrt{3} + 4 i \sqrt{3} - 12}{4}$

Step 1.6.2.3.2

Subtract $12$ from $4$ .

$y = \frac{4 i \sqrt{3} + 4 i \sqrt{3} - 8}{4}$

Step 1.6.2.3.3

Add $4 i \sqrt{3}$ and $4 i \sqrt{3}$ .

$y = \frac{8 i \sqrt{3} - 8}{4}$

Step 1.6.2.4

Reorder $8 i \sqrt{3}$ and $- 8$ .

$y = \frac{- 8 + 8 i \sqrt{3}}{4}$

Step 1.6.2.5

Cancel the common factor of $- 8 + 8 i \sqrt{3}$ and $4$ .

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Step 1.6.2.5.1

Factor $4$ out of $- 8$ .

$y = \frac{4 \cdot - 2 + 8 i \sqrt{3}}{4}$

Step 1.6.2.5.2

Factor $4$ out of $8 i \sqrt{3}$ .

$y = \frac{4 \cdot - 2 + 4 (2 i \sqrt{3})}{4}$

Step 1.6.2.5.3

Factor $4$ out of $4 (- 2) + 4 (2 i \sqrt{3})$ .

$y = \frac{4 (- 2 + 2 i \sqrt{3})}{4}$

Step 1.6.2.5.4

Cancel the common factors.

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Step 1.6.2.5.4.1

Factor $4$ out of $4$ .

$y = \frac{4 (- 2 + 2 i \sqrt{3})}{4 (1)}$

Step 1.6.2.5.4.2

Cancel the common factor.

$y = \frac{4 (- 2 + 2 i \sqrt{3})}{4 \cdot 1}$

Step 1.6.2.5.4.3

Rewrite the expression.

$y = \frac{- 2 + 2 i \sqrt{3}}{1}$

Step 1.6.2.5.4.4

Divide $- 2 + 2 i \sqrt{3}$ by $1$ .

$y = - 2 + 2 i \sqrt{3}$

Step 1.7

List all of the solutions.

$y = 0, x = 0$

$y = 4, x = 4$

$y = - 2 - 2 i \sqrt{3}, x = - 2 + 2 i \sqrt{3}$

$y = - 2 + 2 i \sqrt{3}, x = - 2 - 2 i \sqrt{3}$

$y = 0, x = 0$

$y = 4, x = 4$

$y = - 2 - 2 i \sqrt{3}, x = - 2 + 2 i \sqrt{3}$

$y = - 2 + 2 i \sqrt{3}, x = - 2 - 2 i \sqrt{3}$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3