Calculus Examples

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

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 $2^{2}$ .

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

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 $b = x$ .

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

Step 1.1.3

Simplify the denominator.

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Step 1.1.3.1

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

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

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 $b = x$ .

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

Step 1.1.4

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

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

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

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

Step 1.1.5.2

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

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

Step 1.1.5.3

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

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

Step 1.1.5.4

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

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

Step 1.1.5.5

Add $1$ and $1$ .

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

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

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

Step 1.1.5.6.2

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

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

Step 1.1.5.6.3

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

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

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.

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

Step 1.1.5.6.4.2

Rewrite the expression.

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

Step 1.1.5.6.5

Simplify.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

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.

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

Step 1.5.2.2

Apply the distributive property.

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

Step 1.5.2.3

Apply the distributive property.

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

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 $2$ .

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

Step 1.5.3.1.2

Multiply $- 1$ by $2$ .

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

Step 1.5.3.1.3

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

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

Step 1.5.3.1.4

Rewrite using the commutative property of multiplication.

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

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 $x$ .

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

Step 1.5.3.1.5.2

Multiply $x$ by $x$ .

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

Step 1.5.3.2

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

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

Step 1.5.3.3

Add $4$ and $0$ .

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

Step 1.5.4

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

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

Step 1.5.5

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

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Step 1.5.5.1

Factor $2$ out of $4$ .

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

Step 1.5.5.2

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

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

Step 1.5.5.3

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

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

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

Step 1.6

Move $2$ to the left of $\sqrt{(2 + x) (2 - x)}$ .

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

Step 2

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

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

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.

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

Step 3.1.2

Apply the distributive property.

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

Step 3.1.3

Apply the distributive property.

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

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 $2$ .

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

Step 3.2.1.2

Multiply $- 1$ by $2$ .

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

Step 3.2.1.3

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

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

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

Move $x$ .

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

Step 3.2.1.5.2

Multiply $x$ by $x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Add $4$ and $0$ .

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

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

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Step 5.1

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

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

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

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $2$ .

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

Step 5.2.3.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.3.3

Multiply $2$ by $- 1$ .

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

Step 5.2.3.4

Reorder factors in $4 {(4 - x^{2})}^{\frac{1}{2}} - 2 {(4 - x^{2})}^{\frac{1}{2}} x^{2}$ .

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

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.

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

Step 5.3.1.2

Apply the distributive property.

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

Step 5.3.1.3

Apply the distributive property.

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

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 $2$ .

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

Step 5.3.2.1.2

Multiply $- 1$ by $2$ .

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

Step 5.3.2.1.3

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

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

Step 5.3.2.1.4

Rewrite using the commutative property of multiplication.

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

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 $x$ .

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

Step 5.3.2.1.5.2

Multiply $x$ by $x$ .

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Add $4$ and $0$ .

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

Step 5.3.3

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

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

Step 6

Solve the equation.

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Step 6.1

Factor the left side of the equation.

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Step 6.1.1

Add parentheses.

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

Step 6.1.2

Let $u = {(4 - x^{2})}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(4 - x^{2})}^{\frac{1}{2}}$ .

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

Step 6.1.3

Factor $2 u$ out of $4 u - 2 x^{2} u$ .

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Step 6.1.3.1

Factor $2 u$ out of $4 u$ .

$2 u (2) - 2 x^{2} u = 0$

Step 6.1.3.2

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

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

Step 6.1.3.3

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

$2 u (2 - x^{2}) = 0$

Step 6.1.4

Replace all occurrences of $u$ with ${(4 - x^{2})}^{\frac{1}{2}}$ .

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

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 $0$ .

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

$2 - x^{2} = 0$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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Step 6.3.2.1

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

$4 - x^{2} = 0$

Step 6.3.2.2

Solve for $x$ .

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Step 6.3.2.2.1

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Step 6.3.2.2.2

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

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Step 6.3.2.2.2.1

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

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

Step 6.3.2.2.2.2

Simplify the left side.

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Step 6.3.2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.3.2.2.2.2.2

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

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

Step 6.3.2.2.2.3

Simplify the right side.

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Step 6.3.2.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Step 6.3.2.2.3

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

$x = \pm \sqrt{4}$

Step 6.3.2.2.4

Simplify $\pm \sqrt{4}$ .

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Step 6.3.2.2.4.1

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

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

Step 6.3.2.2.4.2

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

$x = \pm 2$

Step 6.3.2.2.5

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

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Step 6.3.2.2.5.1

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

$x = 2$

Step 6.3.2.2.5.2

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

$x = - 2$

Step 6.3.2.2.5.3

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

$x = 2, - 2$

Step 6.4

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

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Step 6.4.1

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

$2 - x^{2} = 0$

Step 6.4.2

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

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Step 6.4.2.1

Subtract $2$ from both sides of the equation.

$- x^{2} = - 2$

Step 6.4.2.2

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

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Step 6.4.2.2.1

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

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

Step 6.4.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.4.2.2.2.2

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

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

Step 6.4.2.2.3

Simplify the right side.

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Step 6.4.2.2.3.1

Divide $- 2$ by $- 1$ .

$x^{2} = 2$

Step 6.4.2.3

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

$x = \pm \sqrt{2}$

Step 6.4.2.4

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

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Step 6.4.2.4.1

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

$x = \sqrt{2}$

Step 6.4.2.4.2

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

$x = - \sqrt{2}$

Step 6.4.2.4.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 6.5

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

$x = 2, - 2, \sqrt{2}, - \sqrt{2}$

Step 7

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

$x = \sqrt{2}, - \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 1.41421356 \dots, - 1.41421356 \dots$