Calculus Examples

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

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.

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

Step 1.2

Solve $\sqrt{x} = x^{2}$ 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.

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

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

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Step 1.2.2.2.1.1

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

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Step 1.2.2.2.1.1.1

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

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

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.

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

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(x^{2})}^{2}$

Step 1.2.2.2.1.2

Simplify.

$x = {(x^{2})}^{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

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

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Step 1.2.2.3.1.1

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

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

Step 1.2.2.3.1.2

Multiply $2$ by $2$ .

$x = x^{4}$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

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

$x - x^{4} = 0$

Step 1.2.3.2

Factor the left side of the equation.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

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

$x - x^{4} = 0$

Step 1.2.3.2.1.2

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

$x \cdot 1 - x^{4} = 0$

Step 1.2.3.2.1.3

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

$x \cdot 1 + x (- x^{3}) = 0$

Step 1.2.3.2.1.4

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

$x (1 - x^{3}) = 0$

Step 1.2.3.2.2

Rewrite $1$ as $1^{3}$ .

$x (1^{3} - x^{3}) = 0$

Step 1.2.3.2.3

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 $a = 1$ and $b = x$ .

$x ((1 - x) (1^{2} + 1 x + x^{2})) = 0$

Step 1.2.3.2.4

Factor.

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Step 1.2.3.2.4.1

Simplify.

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Step 1.2.3.2.4.1.1

One to any power is one.

$x ((1 - x) (1 + 1 x + x^{2})) = 0$

Step 1.2.3.2.4.1.2

Multiply $x$ by $1$ .

$x ((1 - x) (1 + x + x^{2})) = 0$

Step 1.2.3.2.4.2

Remove unnecessary parentheses.

$x (1 - x) (1 + x + x^{2}) = 0$

Step 1.2.3.3

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

$x = 0$

$1 - x = 0$

$1 + x + x^{2} = 0 + y = x^{2}$

Step 1.2.3.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.5

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.3.5.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 1.2.3.5.2

Solve $1 - x = 0$ for $x$ .

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Step 1.2.3.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.3.5.2.2

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

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Step 1.2.3.5.2.2.1

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

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

Step 1.2.3.5.2.2.2

Simplify the left side.

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Step 1.2.3.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 1.2.3.5.2.2.3

Simplify the right side.

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Step 1.2.3.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.2.3.6

Set $1 + x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.6.1

Set $1 + x + x^{2}$ equal to $0$ .

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

Step 1.2.3.6.2

Solve $1 + x + x^{2} = 0$ for $x$ .

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Step 1.2.3.6.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.3.6.2.2

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

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

Step 1.2.3.6.2.3

Simplify.

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Step 1.2.3.6.2.3.1

Simplify the numerator.

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Step 1.2.3.6.2.3.1.1

One to any power is one.

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

Step 1.2.3.6.2.3.1.2

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

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Step 1.2.3.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.3.1.3

Subtract $4$ from $1$ .

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

Step 1.2.3.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.2.3.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 1.2.3.6.2.3.1.6

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

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

Step 1.2.3.6.2.3.2

Multiply $2$ by $1$ .

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

Step 1.2.3.6.2.4

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

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Step 1.2.3.6.2.4.1

Simplify the numerator.

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Step 1.2.3.6.2.4.1.1

One to any power is one.

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

Step 1.2.3.6.2.4.1.2

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

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Step 1.2.3.6.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.4.1.2.2

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.4.1.3

Subtract $4$ from $1$ .

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

Step 1.2.3.6.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.2.3.6.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 1.2.3.6.2.4.1.6

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

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

Step 1.2.3.6.2.4.2

Multiply $2$ by $1$ .

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

Step 1.2.3.6.2.4.3

Change the $\pm$ to $+$ .

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

Step 1.2.3.6.2.4.4

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

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

Step 1.2.3.6.2.4.5

Factor $- 1$ out of $i \sqrt{3}$ .

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

Step 1.2.3.6.2.4.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

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

Step 1.2.3.6.2.4.7

Move the negative in front of the fraction.

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

Step 1.2.3.6.2.5

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

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Step 1.2.3.6.2.5.1

Simplify the numerator.

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Step 1.2.3.6.2.5.1.1

One to any power is one.

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

Step 1.2.3.6.2.5.1.2

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

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Step 1.2.3.6.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.5.1.2.2

Multiply $- 4$ by $1$ .

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

Step 1.2.3.6.2.5.1.3

Subtract $4$ from $1$ .

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

Step 1.2.3.6.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 1.2.3.6.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 1.2.3.6.2.5.1.6

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

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

Step 1.2.3.6.2.5.2

Multiply $2$ by $1$ .

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

Step 1.2.3.6.2.5.3

Change the $\pm$ to $-$ .

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

Step 1.2.3.6.2.5.4

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

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

Step 1.2.3.6.2.5.5

Factor $- 1$ out of $- i \sqrt{3}$ .

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

Step 1.2.3.6.2.5.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

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

Step 1.2.3.6.2.5.7

Move the negative in front of the fraction.

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

Step 1.2.3.6.2.6

The final answer is the combination of both solutions.

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

Step 1.2.3.7

The final solution is all the values that make $x (1 - x) (1 + x + x^{2}) = 0$ true.

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = {(0)}^{2}$

Step 1.3.2

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

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Step 1.3.2.1

Remove parentheses.

$y = 0^{2}$

Step 1.3.2.2

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

$y = 0$

Step 1.4

Evaluate $y$ when $x = 1$ .

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Step 1.4.1

Substitute $1$ for $x$ .

$y = {(1)}^{2}$

Step 1.4.2

Substitute $1$ for $x$ in $y = {(1)}^{2}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = 1^{2}$

Step 1.4.2.2

One to any power is one.

$y = 1$

Step 1.5

Evaluate $y$ when $x = - \frac{1 - i \sqrt{3}}{2}$ .

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Step 1.5.1

Substitute $- \frac{1 - i \sqrt{3}}{2}$ for $x$ .

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

Step 1.5.2

Simplify ${(- \frac{1 - i \sqrt{3}}{2})}^{2}$ .

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Step 1.5.2.1

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

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Step 1.5.2.1.1

Apply the product rule to $- \frac{1 - i \sqrt{3}}{2}$ .

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

Step 1.5.2.1.2

Apply the product rule to $\frac{1 - i \sqrt{3}}{2}$ .

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

Step 1.5.2.2

Simplify the expression.

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Step 1.5.2.2.1

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

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

Step 1.5.2.2.2

Multiply $\frac{{(1 - i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 1.5.2.2.3

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

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

Step 1.5.2.2.4

Rewrite ${(1 - i \sqrt{3})}^{2}$ as $(1 - i \sqrt{3}) (1 - i \sqrt{3})$ .

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

Step 1.5.2.3

Expand $(1 - i \sqrt{3}) (1 - i \sqrt{3})$ using the FOIL Method.

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Step 1.5.2.3.1

Apply the distributive property.

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

Step 1.5.2.3.2

Apply the distributive property.

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

Step 1.5.2.3.3

Apply the distributive property.

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

Step 1.5.2.4

Simplify and combine like terms.

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Step 1.5.2.4.1

Simplify each term.

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Step 1.5.2.4.1.1

Multiply $1$ by $1$ .

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

Step 1.5.2.4.1.2

Multiply $- i \sqrt{3}$ by $1$ .

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

Step 1.5.2.4.1.3

Multiply $- 1$ by $1$ .

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

Step 1.5.2.4.1.4

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

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Step 1.5.2.4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.2.4.1.4.2

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

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

Step 1.5.2.4.1.4.3

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

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

Step 1.5.2.4.1.4.4

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

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

Step 1.5.2.4.1.4.5

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

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

Step 1.5.2.4.1.4.6

Add $1$ and $1$ .

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

Step 1.5.2.4.1.4.7

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

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

Step 1.5.2.4.1.4.8

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

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

Step 1.5.2.4.1.4.9

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

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

Step 1.5.2.4.1.4.10

Add $1$ and $1$ .

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

Step 1.5.2.4.1.5

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

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Step 1.5.2.4.1.5.1

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

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

Step 1.5.2.4.1.5.2

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

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

Step 1.5.2.4.1.5.3

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

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

Step 1.5.2.4.1.5.4

Cancel the common factor of $2$ .

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Step 1.5.2.4.1.5.4.1

Cancel the common factor.

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

Step 1.5.2.4.1.5.4.2

Rewrite the expression.

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

Step 1.5.2.4.1.5.5

Evaluate the exponent.

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

Step 1.5.2.4.1.6

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

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

Step 1.5.2.4.1.7

Multiply $3$ by $- 1$ .

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

Step 1.5.2.4.2

Subtract $3$ from $1$ .

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

Step 1.5.2.4.3

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

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

Step 1.5.2.5

Reorder $- 2 i \sqrt{3}$ and $- 2$ .

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

Step 1.5.2.6

Cancel the common factor of $- 2 - 2 i \sqrt{3}$ and $4$ .

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Step 1.5.2.6.1

Factor $2$ out of $- 2$ .

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

Step 1.5.2.6.2

Factor $2$ out of $- 2 i \sqrt{3}$ .

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

Step 1.5.2.6.3

Factor $2$ out of $2 (- 1) + 2 (- i \sqrt{3})$ .

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

Step 1.5.2.6.4

Cancel the common factors.

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Step 1.5.2.6.4.1

Factor $2$ out of $4$ .

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

Step 1.5.2.6.4.2

Cancel the common factor.

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

Step 1.5.2.6.4.3

Rewrite the expression.

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

Step 1.5.2.7

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

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

Step 1.5.2.8

Factor $- 1$ out of $- i \sqrt{3}$ .

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

Step 1.5.2.9

Factor $- 1$ out of $- 1 (1) - (i \sqrt{3})$ .

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

Step 1.5.2.10

Move the negative in front of the fraction.

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

Step 1.6

Evaluate $y$ when $x = - \frac{1 + i \sqrt{3}}{2}$ .

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Step 1.6.1

Substitute $- \frac{1 + i \sqrt{3}}{2}$ for $x$ .

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

Step 1.6.2

Simplify ${(- \frac{1 + i \sqrt{3}}{2})}^{2}$ .

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Step 1.6.2.1

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

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Step 1.6.2.1.1

Apply the product rule to $- \frac{1 + i \sqrt{3}}{2}$ .

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

Step 1.6.2.1.2

Apply the product rule to $\frac{1 + i \sqrt{3}}{2}$ .

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

Step 1.6.2.2

Simplify the expression.

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Step 1.6.2.2.1

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

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

Step 1.6.2.2.2

Multiply $\frac{{(1 + i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

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

Step 1.6.2.2.3

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

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

Step 1.6.2.2.4

Rewrite ${(1 + i \sqrt{3})}^{2}$ as $(1 + i \sqrt{3}) (1 + i \sqrt{3})$ .

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

Step 1.6.2.3

Expand $(1 + i \sqrt{3}) (1 + i \sqrt{3})$ using the FOIL Method.

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Step 1.6.2.3.1

Apply the distributive property.

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

Step 1.6.2.3.2

Apply the distributive property.

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

Step 1.6.2.3.3

Apply the distributive property.

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

Step 1.6.2.4

Simplify and combine like terms.

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Step 1.6.2.4.1

Simplify each term.

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Step 1.6.2.4.1.1

Multiply $1$ by $1$ .

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

Step 1.6.2.4.1.2

Multiply $i \sqrt{3}$ by $1$ .

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

Step 1.6.2.4.1.3

Multiply $i$ by $1$ .

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

Step 1.6.2.4.1.4

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

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Step 1.6.2.4.1.4.1

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

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

Step 1.6.2.4.1.4.2

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

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

Step 1.6.2.4.1.4.3

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

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

Step 1.6.2.4.1.4.4

Add $1$ and $1$ .

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

Step 1.6.2.4.1.4.5

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

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

Step 1.6.2.4.1.4.6

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

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

Step 1.6.2.4.1.4.7

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

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

Step 1.6.2.4.1.4.8

Add $1$ and $1$ .

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

Step 1.6.2.4.1.5

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

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

Step 1.6.2.4.1.6

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

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Step 1.6.2.4.1.6.1

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

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

Step 1.6.2.4.1.6.2

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

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

Step 1.6.2.4.1.6.3

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

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

Step 1.6.2.4.1.6.4

Cancel the common factor of $2$ .

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Step 1.6.2.4.1.6.4.1

Cancel the common factor.

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

Step 1.6.2.4.1.6.4.2

Rewrite the expression.

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

Step 1.6.2.4.1.6.5

Evaluate the exponent.

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

Step 1.6.2.4.1.7

Multiply $- 1$ by $3$ .

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

Step 1.6.2.4.2

Subtract $3$ from $1$ .

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

Step 1.6.2.4.3

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

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

Step 1.6.2.5

Reorder $2 i \sqrt{3}$ and $- 2$ .

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

Step 1.6.2.6

Cancel the common factor of $- 2 + 2 i \sqrt{3}$ and $4$ .

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Step 1.6.2.6.1

Factor $2$ out of $- 2$ .

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

Step 1.6.2.6.2

Factor $2$ out of $2 i \sqrt{3}$ .

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

Step 1.6.2.6.3

Factor $2$ out of $2 \cdot - 1 + 2 (i \sqrt{3})$ .

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

Step 1.6.2.6.4

Cancel the common factors.

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Step 1.6.2.6.4.1

Factor $2$ out of $4$ .

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

Step 1.6.2.6.4.2

Cancel the common factor.

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

Step 1.6.2.6.4.3

Rewrite the expression.

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

Step 1.6.2.7

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

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

Step 1.6.2.8

Factor $- 1$ out of $i \sqrt{3}$ .

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

Step 1.6.2.9

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{3})$ .

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

Step 1.6.2.10

Move the negative in front of the fraction.

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

Step 1.7

List all of the solutions.

$y = 0, x = 0$

$y = 1, x = 1$

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

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

$y = 0, x = 0$

$y = 1, x = 1$

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

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

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3