Calculus Examples

$x \sqrt{4 - x^{2}} - \frac{x^{2}}{\sqrt{4 - x^{2}}} = 0$

Step 1

Simplify both sides of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$x \sqrt{2^{2} - x^{2}} - \frac{x^{2}}{\sqrt{4 - x^{2}}} = 0$

Step 1.1.2

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 $b = x$ .

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

Step 1.1.3

Simplify the denominator.

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Step 1.1.3.1

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

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2}}{\sqrt{2^{2} - x^{2}}} = 0$

Step 1.1.3.2

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 $b = x$ .

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2}}{\sqrt{(2 + x) (2 - x)}} = 0$

Step 1.1.4

Multiply $\frac{x^{2}}{\sqrt{(2 + x) (2 - x)}}$ by $\frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}$ .

$x \sqrt{(2 + x) (2 - x)} - (\frac{x^{2}}{\sqrt{(2 + x) (2 - x)}} \cdot \frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}) = 0$

Step 1.1.5

Combine and simplify the denominator.

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Step 1.1.5.1

Multiply $\frac{x^{2}}{\sqrt{(2 + x) (2 - x)}}$ by $\frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}$ .

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)} \sqrt{(2 + x) (2 - x)}} = 0$

Step 1.1.5.2

Raise $\sqrt{(2 + x) (2 - x)}$ to the power of $1$ .

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

Step 1.1.5.3

Raise $\sqrt{(2 + x) (2 - x)}$ to the power of $1$ .

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

Step 1.1.5.4

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

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

Step 1.1.5.5

Add $1$ and $1$ .

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{{\sqrt{(2 + x) (2 - x)}}^{2}} = 0$

Step 1.1.5.6

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

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Step 1.1.5.6.1

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

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

Step 1.1.5.6.2

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

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{1}{2} \cdot 2}} = 0$

Step 1.1.5.6.3

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

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{2}{2}}} = 0$

Step 1.1.5.6.4

Cancel the common factor of $2$ .

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Step 1.1.5.6.4.1

Cancel the common factor.

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{2}{2}}} = 0$

Step 1.1.5.6.4.2

Rewrite the expression.

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

Step 1.1.5.6.5

Simplify.

$x \sqrt{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{(2 + x) (2 - x)} = 0$

Step 1.2

To write $x \sqrt{(2 + x) (2 - x)}$ as a fraction with a common denominator, multiply by $\frac{(2 + x) (2 - x)}{(2 + x) (2 - x)}$ .

$x \sqrt{(2 + x) (2 - x)} \cdot \frac{(2 + x) (2 - x)}{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{(2 + x) (2 - x)} = 0$

Step 1.3

Combine $x \sqrt{(2 + x) (2 - x)}$ and $\frac{(2 + x) (2 - x)}{(2 + x) (2 - x)}$ .

$\frac{x \sqrt{(2 + x) (2 - x)} ((2 + x) (2 - x))}{(2 + x) (2 - x)} - \frac{x^{2} \sqrt{(2 + x) (2 - x)}}{(2 + x) (2 - x)} = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x \sqrt{(2 + x) (2 - x)} ((2 + x) (2 - x)) - x^{2} \sqrt{(2 + x) (2 - x)}}{(2 + x) (2 - x)} = 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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Step 1.5.1.1

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

$\frac{x \sqrt{(2 + x) (2 - x)} ((2 + x) (2 - x)) + x \sqrt{(2 + x) (2 - x)} (- x)}{(2 + x) (2 - x)} = 0$

Step 1.5.1.2

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

$\frac{x \sqrt{(2 + x) (2 - x)} ((2 + x) (2 - x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.2

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 1.5.2.1

Apply the distributive property.

$\frac{x \sqrt{(2 + x) (2 - x)} (2 (2 - x) + x (2 - x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.2.2

Apply the distributive property.

$\frac{x \sqrt{(2 + x) (2 - x)} (2 \cdot 2 + 2 (- x) + x (2 - x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.2.3

Apply the distributive property.

$\frac{x \sqrt{(2 + x) (2 - x)} (2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3

Simplify and combine like terms.

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Step 1.5.3.1

Simplify each term.

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Step 1.5.3.1.1

Multiply $2$ by $2$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (4 + 2 (- x) + x \cdot 2 + x (- x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3.1.2

Multiply $- 1$ by $2$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (4 - 2 x + x \cdot 2 + x (- x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3.1.3

Move $2$ to the left of $x$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (4 - 2 x + 2 \cdot x + x (- x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{x \sqrt{(2 + x) (2 - x)} (4 - 2 x + 2 x - x \cdot x - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.3.1.5.1

Move $x$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (4 - 2 x + 2 x - (x \cdot x) - x)}{(2 + x) (2 - x)} = 0$

Step 1.5.3.1.5.2

Multiply $x$ by $x$ .

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

Step 1.5.3.2

Add $- 2 x$ and $2 x$ .

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

Step 1.5.3.3

Add $4$ and $0$ .

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

Step 1.5.4

Reorder terms.

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

Step 1.6

Factor $- 1$ out of $- x^{2}$ .

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

Step 1.7

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

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

Step 1.8

Factor $- 1$ out of $- (x^{2}) - (x)$ .

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

Step 1.9

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (- (x^{2} + x) - 1 (- 4))}{(2 + x) (2 - x)} = 0$

Step 1.10

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 4)$ .

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

Step 1.11

Rewrite $- (x^{2} + x - 4)$ as $- 1 (x^{2} + x - 4)$ .

$\frac{x \sqrt{(2 + x) (2 - x)} (- 1 (x^{2} + x - 4))}{(2 + x) (2 - x)} = 0$

Step 1.12

Move the negative in front of the fraction.

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

Step 1.13

Reorder factors in $- \frac{(x \sqrt{(2 + x) (2 - x)}) (x^{2} + x - 4)}{(2 + x) (2 - x)}$ .

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

Step 2

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

$- \frac{x {((2 + x) (2 - x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3

Simplify the numerator.

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Step 3.1

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$- \frac{x {(2 (2 - x) + x (2 - x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.1.2

Apply the distributive property.

$- \frac{x {(2 \cdot 2 + 2 (- x) + x (2 - x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.1.3

Apply the distributive property.

$- \frac{x {(2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $2$ by $2$ .

$- \frac{x {(4 + 2 (- x) + x \cdot 2 + x (- x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.1.2

Multiply $- 1$ by $2$ .

$- \frac{x {(4 - 2 x + x \cdot 2 + x (- x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.1.3

Move $2$ to the left of $x$ .

$- \frac{x {(4 - 2 x + 2 x + x (- x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

$- \frac{x {(4 - 2 x + 2 x - x \cdot x)}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.5.1

Move $x$ .

$- \frac{x {(4 - 2 x + 2 x - (x \cdot x))}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.1.5.2

Multiply $x$ by $x$ .

$- \frac{x {(4 - 2 x + 2 x - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.2

Add $- 2 x$ and $2 x$ .

$- \frac{x {(4 + 0 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 3.2.3

Add $4$ and $0$ .

$- \frac{x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(2 + x) (2 - x), 1$

Step 4.2

The LCM of one and any expression is the expression.

$(2 + x) (2 - x)$

Step 5

Multiply each term in $- \frac{x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$ by $(2 + x) (2 - x)$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} = 0$ by $(2 + x) (2 - x)$ .

$- \frac{x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} ((2 + x) (2 - x)) = 0 ((2 + x) (2 - x))$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $(2 + x) (2 - x)$ .

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Step 5.2.1.1

Move the leading negative in $- \frac{x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)}$ into the numerator.

$\frac{- x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} ((2 + x) (2 - x)) = 0 ((2 + x) (2 - x))$

Step 5.2.1.2

Cancel the common factor.

$\frac{- x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4)}{(2 + x) (2 - x)} ((2 + x) (2 - x)) = 0 ((2 + x) (2 - x))$

Step 5.2.1.3

Rewrite the expression.

$- x {(4 - x^{2})}^{\frac{1}{2}} (x^{2} + x - 4) = 0 ((2 + x) (2 - x))$

Step 5.2.2

Apply the distributive property.

$- x {(4 - x^{2})}^{\frac{1}{2}} x^{2} - x {(4 - x^{2})}^{\frac{1}{2}} x - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.2.3.1.1

Move $x^{2}$ .

$- (x^{2} x) {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} x - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 5.2.3.1.2.1

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

$- (x^{2} x^{1}) {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} x - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.1.2.2

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

$- x^{2 + 1} {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} x - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.1.3

Add $2$ and $1$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} x - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.3.2.1

Move $x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - (x \cdot x) {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.2.2

Multiply $x$ by $x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} - x {(4 - x^{2})}^{\frac{1}{2}} \cdot - 4 = 0 ((2 + x) (2 - x))$

Step 5.2.3.3

Multiply $- 4$ by $- 1$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 ((2 + x) (2 - x))$

Step 5.3

Simplify the right side.

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Step 5.3.1

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 5.3.1.1

Apply the distributive property.

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (2 (2 - x) + x (2 - x))$

Step 5.3.1.2

Apply the distributive property.

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (2 \cdot 2 + 2 (- x) + x (2 - x))$

Step 5.3.1.3

Apply the distributive property.

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x))$

Step 5.3.2

Simplify and combine like terms.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $2$ by $2$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 + 2 (- x) + x \cdot 2 + x (- x))$

Step 5.3.2.1.2

Multiply $- 1$ by $2$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - 2 x + x \cdot 2 + x (- x))$

Step 5.3.2.1.3

Move $2$ to the left of $x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - 2 x + 2 \cdot x + x (- x))$

Step 5.3.2.1.4

Rewrite using the commutative property of multiplication.

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - 2 x + 2 x - x \cdot x)$

Step 5.3.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.2.1.5.1

Move $x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - 2 x + 2 x - (x \cdot x))$

Step 5.3.2.1.5.2

Multiply $x$ by $x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - 2 x + 2 x - x^{2})$

Step 5.3.2.2

Add $- 2 x$ and $2 x$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 + 0 - x^{2})$

Step 5.3.2.3

Add $4$ and $0$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0 (4 - x^{2})$

Step 5.3.3

Multiply $0$ by $4 - x^{2}$ .

$- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0$

Step 6

Solve the equation.

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Step 6.1

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $- x^{3} {(4 - x^{2})}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}}$ .

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Step 6.1.1

Reorder the expression.

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Step 6.1.1.1

Reorder $4$ and $- x^{2}$ .

$- x^{3} {(- x^{2} + 4)}^{\frac{1}{2}} - x^{2} {(4 - x^{2})}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0$

Step 6.1.1.2

Reorder $4$ and $- x^{2}$ .

$- x^{3} {(- x^{2} + 4)}^{\frac{1}{2}} - x^{2} {(- x^{2} + 4)}^{\frac{1}{2}} + 4 x {(4 - x^{2})}^{\frac{1}{2}} = 0$

Step 6.1.1.3

Reorder $4$ and $- x^{2}$ .

$- x^{3} {(- x^{2} + 4)}^{\frac{1}{2}} - x^{2} {(- x^{2} + 4)}^{\frac{1}{2}} + 4 x {(- x^{2} + 4)}^{\frac{1}{2}} = 0$

Step 6.1.2

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $- x^{3} {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$- x {(- x^{2} + 4)}^{\frac{1}{2}} x^{2} - x^{2} {(- x^{2} + 4)}^{\frac{1}{2}} + 4 x {(- x^{2} + 4)}^{\frac{1}{2}} = 0$

Step 6.1.3

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $- x^{2} {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$- x {(- x^{2} + 4)}^{\frac{1}{2}} x^{2} - x {(- x^{2} + 4)}^{\frac{1}{2}} x + 4 x {(- x^{2} + 4)}^{\frac{1}{2}} = 0$

Step 6.1.4

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $4 x {(- x^{2} + 4)}^{\frac{1}{2}}$ .

$- x {(- x^{2} + 4)}^{\frac{1}{2}} x^{2} - x {(- x^{2} + 4)}^{\frac{1}{2}} x - x {(- x^{2} + 4)}^{\frac{1}{2}} \cdot - 4 = 0$

Step 6.1.5

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $- x {(- x^{2} + 4)}^{\frac{1}{2}} (x^{2}) - x {(- x^{2} + 4)}^{\frac{1}{2}} (x)$ .

$- x {(- x^{2} + 4)}^{\frac{1}{2}} (x^{2} + x) - x {(- x^{2} + 4)}^{\frac{1}{2}} \cdot - 4 = 0$

Step 6.1.6

Factor $- x {(- x^{2} + 4)}^{\frac{1}{2}}$ out of $- x {(- x^{2} + 4)}^{\frac{1}{2}} (x^{2} + x) - x {(- x^{2} + 4)}^{\frac{1}{2}} (- 4)$ .

$- x {(- x^{2} + 4)}^{\frac{1}{2}} (x^{2} + x - 4) = 0$

Step 6.2

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

$x = 0$

${(- x^{2} + 4)}^{\frac{1}{2}} = 0$

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

Step 6.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.4

Set ${(- x^{2} + 4)}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set ${(- x^{2} + 4)}^{\frac{1}{2}}$ equal to $0$ .

${(- x^{2} + 4)}^{\frac{1}{2}} = 0$

Step 6.4.2

Solve ${(- x^{2} + 4)}^{\frac{1}{2}} = 0$ for $x$ .

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Step 6.4.2.1

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

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

Step 6.4.2.2

Solve for $x$ .

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Step 6.4.2.2.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 6.4.2.2.2

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

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Step 6.4.2.2.2.1

Divide each term in $- x^{2} = - 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Step 6.4.2.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Step 6.4.2.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 1}$

Step 6.4.2.2.2.3

Simplify the right side.

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Step 6.4.2.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 6.4.2.2.3

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

$x = \pm \sqrt{4}$

Step 6.4.2.2.4

Simplify $\pm \sqrt{4}$ .

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Step 6.4.2.2.4.1

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

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

Step 6.4.2.2.4.2

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

$x = \pm 2$

Step 6.4.2.2.5

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

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Step 6.4.2.2.5.1

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

$x = 2$

Step 6.4.2.2.5.2

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

$x = - 2$

Step 6.4.2.2.5.3

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

$x = 2, - 2$

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

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

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Step 6.5.2.1

Use the quadratic formula to find the solutions.

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

Step 6.5.2.2

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

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

Step 6.5.2.3

Simplify.

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Step 6.5.2.3.1

Simplify the numerator.

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Step 6.5.2.3.1.1

One to any power is one.

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

Step 6.5.2.3.1.2

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

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Step 6.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.5.2.3.1.2.2

Multiply $- 4$ by $- 4$ .

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

Step 6.5.2.3.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot 1}$

Step 6.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{17}}{2}$

Step 6.5.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{17}}{2}, - \frac{1 + \sqrt{17}}{2}$

Step 6.6

The final solution is all the values that make $- x {(- x^{2} + 4)}^{\frac{1}{2}} (x^{2} + x - 4) = 0$ true.

$x = 0, 2, - 2, - \frac{1 - \sqrt{17}}{2}, - \frac{1 + \sqrt{17}}{2}$

Step 7

Exclude the solutions that do not make $x \sqrt{4 - x^{2}} - \frac{x^{2}}{\sqrt{4 - x^{2}}} = 0$ true.

$x = 0, - \frac{1 - \sqrt{17}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{1 - \sqrt{17}}{2}$

Decimal Form:

$x = 0, 1.56155281 \dots$