Calculus Examples

$x^{2} + x y + y^{2} = 3$

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Subtract $3$ from both sides of the equation.

$x^{2} + x y + y^{2} - 3 = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 1$ , $b = x$ , and $c = x^{2} - 3$ into the quadratic formula and solve for $y$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Multiply $- 4$ by $1$ .

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

Step 1.4.1.2

Apply the distributive property.

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

Step 1.4.1.3

Multiply $- 4$ by $- 3$ .

$y = \frac{- x \pm \sqrt{x^{2} - 4 x^{2} + 12}}{2 \cdot 1}$

Step 1.4.1.4

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{- x \pm \sqrt{- 3 x^{2} + 12}}{2 \cdot 1}$

Step 1.4.1.5

Rewrite $- 3 x^{2} + 12$ in a factored form.

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Step 1.4.1.5.1

Factor $3$ out of $- 3 x^{2} + 12$ .

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Step 1.4.1.5.1.1

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

$y = \frac{- x \pm \sqrt{3 (- x^{2}) + 12}}{2 \cdot 1}$

Step 1.4.1.5.1.2

Factor $3$ out of $12$ .

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

Step 1.4.1.5.1.3

Factor $3$ out of $3 (- x^{2}) + 3 (4)$ .

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

Step 1.4.1.5.2

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

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

Step 1.4.1.5.3

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

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

Step 1.4.1.5.4

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

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

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{- x \pm \sqrt{3 (2 + x) (2 - x)}}{2}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Multiply $- 4$ by $1$ .

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

Step 1.5.1.2

Apply the distributive property.

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

Step 1.5.1.3

Multiply $- 4$ by $- 3$ .

$y = \frac{- x \pm \sqrt{x^{2} - 4 x^{2} + 12}}{2 \cdot 1}$

Step 1.5.1.4

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{- x \pm \sqrt{- 3 x^{2} + 12}}{2 \cdot 1}$

Step 1.5.1.5

Rewrite $- 3 x^{2} + 12$ in a factored form.

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Step 1.5.1.5.1

Factor $3$ out of $- 3 x^{2} + 12$ .

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Step 1.5.1.5.1.1

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

$y = \frac{- x \pm \sqrt{3 (- x^{2}) + 12}}{2 \cdot 1}$

Step 1.5.1.5.1.2

Factor $3$ out of $12$ .

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

Step 1.5.1.5.1.3

Factor $3$ out of $3 (- x^{2}) + 3 (4)$ .

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

Step 1.5.1.5.2

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

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

Step 1.5.1.5.3

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

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

Step 1.5.1.5.4

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

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

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{- x \pm \sqrt{3 (2 + x) (2 - x)}}{2}$

Step 1.5.3

Change the $\pm$ to $+$ .

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

Step 1.5.4

Factor $- 1$ out of $- x$ .

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

Rewrite $- (x - \sqrt{3 (2 + x) (2 - x)})$ as $- 1 (x - \sqrt{3 (2 + x) (2 - x)})$ .

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

Step 1.5.8

Move the negative in front of the fraction.

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

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Multiply $- 4$ by $1$ .

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

Step 1.6.1.2

Apply the distributive property.

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

Step 1.6.1.3

Multiply $- 4$ by $- 3$ .

$y = \frac{- x \pm \sqrt{x^{2} - 4 x^{2} + 12}}{2 \cdot 1}$

Step 1.6.1.4

Subtract $4 x^{2}$ from $x^{2}$ .

$y = \frac{- x \pm \sqrt{- 3 x^{2} + 12}}{2 \cdot 1}$

Step 1.6.1.5

Rewrite $- 3 x^{2} + 12$ in a factored form.

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Step 1.6.1.5.1

Factor $3$ out of $- 3 x^{2} + 12$ .

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Step 1.6.1.5.1.1

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

$y = \frac{- x \pm \sqrt{3 (- x^{2}) + 12}}{2 \cdot 1}$

Step 1.6.1.5.1.2

Factor $3$ out of $12$ .

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

Step 1.6.1.5.1.3

Factor $3$ out of $3 (- x^{2}) + 3 (4)$ .

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

Step 1.6.1.5.2

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

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

Step 1.6.1.5.3

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

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

Step 1.6.1.5.4

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

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

Step 1.6.2

Multiply $2$ by $1$ .

$y = \frac{- x \pm \sqrt{3 (2 + x) (2 - x)}}{2}$

Step 1.6.3

Change the $\pm$ to $-$ .

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

Step 1.6.4

Factor $- 1$ out of $- x$ .

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

Step 1.6.5

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

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

Step 1.6.6

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

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

Step 1.6.7

Rewrite $- (x + \sqrt{3 (2 + x) (2 - x)})$ as $- 1 (x + \sqrt{3 (2 + x) (2 - x)})$ .

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

Step 1.6.8

Move the negative in front of the fraction.

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

Step 1.7

The final answer is the combination of both solutions.

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

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

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

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

Step 2

Set each solution of $y$ as a function of $x$ .

$y = - \frac{x - \sqrt{3 (2 + x) (2 - x)}}{2} \to f (x) = - \frac{x - \sqrt{3 (2 + x) (2 - x)}}{2}$

$y = - \frac{x + \sqrt{3 (2 + x) (2 - x)}}{2} \to f (x) = - \frac{x + \sqrt{3 (2 + x) (2 - x)}}{2}$

Step 3

Because the $y$ variable in the equation $x^{2} + x y + y^{2} = 3$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + x y + y^{2}) = \frac{d}{d x} (3)$

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

Differentiate.

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Step 3.2.1.1

By the Sum Rule, the derivative of $x^{2} + x y + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x y] + \frac{d}{d x} [y^{2}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x y] + \frac{d}{d x} [y^{2}]$

Step 3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [x y] + \frac{d}{d x} [y^{2}]$

Step 3.2.2

Evaluate $\frac{d}{d x} [x y]$ .

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Step 3.2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$2 x + x \frac{d}{d x} [y] + y \frac{d}{d x} [x] + \frac{d}{d x} [y^{2}]$

Step 3.2.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x + x y' + y \frac{d}{d x} [x] + \frac{d}{d x} [y^{2}]$

Step 3.2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 x + x y' + y \cdot 1 + \frac{d}{d x} [y^{2}]$

Step 3.2.2.4

Multiply $y$ by $1$ .

$2 x + x y' + y + \frac{d}{d x} [y^{2}]$

Step 3.2.3

Evaluate $\frac{d}{d x} [y^{2}]$ .

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Step 3.2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.3.1.1

To apply the Chain Rule, set $u$ as $y$ .

$2 x + x y' + y + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y]$

Step 3.2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 x + x y' + y + 2 u \frac{d}{d x} [y]$

Step 3.2.3.1.3

Replace all occurrences of $u$ with $y$ .

$2 x + x y' + y + 2 y \frac{d}{d x} [y]$

Step 3.2.3.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x + x y' + y + 2 y y'$

Step 3.2.4

Reorder terms.

$x y' + 2 y y' + 2 x + y$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0$

Step 3.4

Reform the equation by setting the left side equal to the right side.

$x y' + 2 y y' + 2 x + y = 0$

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 3.5.1.1

Subtract $2 x$ from both sides of the equation.

$x y' + 2 y y' + y = - 2 x$

Step 3.5.1.2

Subtract $y$ from both sides of the equation.

$x y' + 2 y y' = - 2 x - y$

Step 3.5.2

Factor $y'$ out of $x y' + 2 y y'$ .

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Step 3.5.2.1

Factor $y'$ out of $x y'$ .

$y' x + 2 y y' = - 2 x - y$

Step 3.5.2.2

Factor $y'$ out of $2 y y'$ .

$y' (x) + y' (2 y) = - 2 x - y$

Step 3.5.2.3

Factor $y'$ out of $y' (x) + y' (2 y)$ .

$y' (x + 2 y) = - 2 x - y$

Step 3.5.3

Divide each term in $y' (x + 2 y) = - 2 x - y$ by $x + 2 y$ and simplify.

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Step 3.5.3.1

Divide each term in $y' (x + 2 y) = - 2 x - y$ by $x + 2 y$ .

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 2 x}{x + 2 y} + \frac{- y}{x + 2 y}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x + 2 y$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 2 x}{x + 2 y} + \frac{- y}{x + 2 y}$

Step 3.5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{x + 2 y} + \frac{- y}{x + 2 y}$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{- 2 x - y}{x + 2 y}$

Step 3.5.3.3.2

Factor $- 1$ out of $- 2 x$ .

$y' = \frac{- (2 x) - y}{x + 2 y}$

Step 3.5.3.3.3

Factor $- 1$ out of $- y$ .

$y' = \frac{- (2 x) - (y)}{x + 2 y}$

Step 3.5.3.3.4

Factor $- 1$ out of $- (2 x) - (y)$ .

$y' = \frac{- (2 x + y)}{x + 2 y}$

Step 3.5.3.3.5

Simplify the expression.

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Step 3.5.3.3.5.1

Rewrite $- (2 x + y)$ as $- 1 (2 x + y)$ .

$y' = \frac{- 1 (2 x + y)}{x + 2 y}$

Step 3.5.3.3.5.2

Move the negative in front of the fraction.

$y' = - \frac{2 x + y}{x + 2 y}$

Step 3.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{2 x + y}{x + 2 y}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{2 x + y}{x + 2 y} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$2 x + y = 0$

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Subtract $y$ from both sides of the equation.

$2 x = - y$

Step 4.2.2

Divide each term in $2 x = - y$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 x = - y$ by $2$ .

$\frac{2 x}{2} = \frac{- y}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- y}{2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y}{2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{y}{2}$

Step 5

Solve the function $f (x) = - \frac{x - \sqrt{3 (2 + x) (2 - x)}}{2}$ at $x = - \frac{y}{2}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{y}{2}$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

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

Step 5.2.2

Simplify the numerator.

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Step 5.2.2.1

Combine exponents.

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Step 5.2.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.2.1.2

Multiply $\frac{y}{2}$ by $1$ .

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

Step 5.2.2.2

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

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

Step 5.2.2.3

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

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

Step 5.2.2.4

Combine the numerators over the common denominator.

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

Step 5.2.2.5

Multiply $2$ by $2$ .

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

Step 5.2.2.6

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

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

Step 5.2.2.7

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

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

Step 5.2.2.8

Combine the numerators over the common denominator.

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

Step 5.2.2.9

Multiply $2$ by $2$ .

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

Step 5.2.2.10

Combine exponents.

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Step 5.2.2.10.1

Combine $3$ and $\frac{4 - y}{2}$ .

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

Step 5.2.2.10.2

Multiply $\frac{3 (4 - y)}{2}$ by $\frac{4 + y}{2}$ .

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

Step 5.2.2.10.3

Multiply $2$ by $2$ .

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

Step 5.2.2.11

Rewrite $\frac{3 (4 - y) (4 + y)}{4}$ as ${(\frac{1}{2})}^{2} (3 (4 - y) (4 + y))$ .

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Step 5.2.2.11.1

Factor the perfect power $1^{2}$ out of $3 (4 - y) (4 + y)$ .

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

Step 5.2.2.11.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 5.2.2.11.3

Rearrange the fraction $\frac{1^{2} (3 (4 - y) (4 + y))}{2^{2} \cdot 1}$ .

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

Step 5.2.2.12

Pull terms out from under the radical.

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

Step 5.2.2.13

Combine $\frac{1}{2}$ and $\sqrt{3 (4 - y) (4 + y)}$ .

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

Step 5.2.2.14

Combine the numerators over the common denominator.

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

Step 5.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.4

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

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Step 5.2.4.1

Multiply $\frac{- y - \sqrt{3 (4 - y) (4 + y)}}{2}$ by $\frac{1}{2}$ .

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

Step 5.2.4.2

Multiply $2$ by $2$ .

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

Step 5.2.5

Factor $- 1$ out of $- y$ .

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

Step 5.2.6

Factor $- 1$ out of $- \sqrt{3 (4 - y) (4 + y)}$ .

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

Step 5.2.7

Factor $- 1$ out of $- (y) - (\sqrt{3 (4 - y) (4 + y)})$ .

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

Step 5.2.8

Simplify the expression.

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Step 5.2.8.1

Rewrite $- (y + \sqrt{3 (4 - y) (4 + y)})$ as $- 1 (y + \sqrt{3 (4 - y) (4 + y)})$ .

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

Step 5.2.8.2

Move the negative in front of the fraction.

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

Step 5.2.8.3

Multiply $- 1$ by $- 1$ .

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

Step 5.2.8.4

Multiply $\frac{y + \sqrt{3 (4 - y) (4 + y)}}{4}$ by $1$ .

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

Step 5.2.9

The final answer is $\frac{y + \sqrt{3 (4 - y) (4 + y)}}{4}$ .

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

Step 6

Solve the function $f (x) = - \frac{x + \sqrt{3 (2 + x) (2 - x)}}{2}$ at $x = - \frac{y}{2}$ .

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Step 6.1

Replace the variable $x$ with $- \frac{y}{2}$ in the expression.

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

Step 6.2

Simplify the result.

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Step 6.2.1

Remove parentheses.

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

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

Combine exponents.

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Step 6.2.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.2.1.2

Multiply $\frac{y}{2}$ by $1$ .

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

Step 6.2.2.2

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

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

Step 6.2.2.3

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

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

Step 6.2.2.4

Combine the numerators over the common denominator.

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

Step 6.2.2.5

Multiply $2$ by $2$ .

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

Step 6.2.2.6

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

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

Step 6.2.2.7

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

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

Step 6.2.2.8

Combine the numerators over the common denominator.

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

Step 6.2.2.9

Multiply $2$ by $2$ .

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

Step 6.2.2.10

Combine exponents.

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Step 6.2.2.10.1

Combine $3$ and $\frac{4 - y}{2}$ .

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

Step 6.2.2.10.2

Multiply $\frac{3 (4 - y)}{2}$ by $\frac{4 + y}{2}$ .

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

Step 6.2.2.10.3

Multiply $2$ by $2$ .

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

Step 6.2.2.11

Rewrite $\frac{3 (4 - y) (4 + y)}{4}$ as ${(\frac{1}{2})}^{2} (3 (4 - y) (4 + y))$ .

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Step 6.2.2.11.1

Factor the perfect power $1^{2}$ out of $3 (4 - y) (4 + y)$ .

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

Step 6.2.2.11.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 6.2.2.11.3

Rearrange the fraction $\frac{1^{2} (3 (4 - y) (4 + y))}{2^{2} \cdot 1}$ .

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

Step 6.2.2.12

Pull terms out from under the radical.

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

Step 6.2.2.13

Combine $\frac{1}{2}$ and $\sqrt{3 (4 - y) (4 + y)}$ .

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

Step 6.2.2.14

Combine the numerators over the common denominator.

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

Step 6.2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2.4

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

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Step 6.2.4.1

Multiply $\frac{- y + \sqrt{3 (4 - y) (4 + y)}}{2}$ by $\frac{1}{2}$ .

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

Step 6.2.4.2

Multiply $2$ by $2$ .

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

Step 6.2.5

Factor $- 1$ out of $- y$ .

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

Step 6.2.6

Factor $- 1$ out of $\sqrt{3 (4 - y) (4 + y)}$ .

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

Step 6.2.7

Factor $- 1$ out of $- (y) - 1 (- \sqrt{3 (4 - y) (4 + y)})$ .

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

Step 6.2.8

Simplify the expression.

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Step 6.2.8.1

Rewrite $- (y - \sqrt{3 (4 - y) (4 + y)})$ as $- 1 (y - \sqrt{3 (4 - y) (4 + y)})$ .

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

Step 6.2.8.2

Move the negative in front of the fraction.

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

Step 6.2.8.3

Multiply $- 1$ by $- 1$ .

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

Step 6.2.8.4

Multiply $\frac{y - \sqrt{3 (4 - y) (4 + y)}}{4}$ by $1$ .

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

Step 6.2.9

The final answer is $\frac{y - \sqrt{3 (4 - y) (4 + y)}}{4}$ .

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

Step 7

The horizontal tangent lines are $y = \frac{y + \sqrt{3 (4 - y) (4 + y)}}{4}, y = \frac{y - \sqrt{3 (4 - y) (4 + y)}}{4}$

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

Step 8