Calculus Examples

$\sqrt{25 - x^{2}} - \frac{x^{2}}{\sqrt{25 - x^{2}}} = 0$

Step 1

Simplify both sides of the equation.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} - x^{2}} - \frac{x^{2}}{\sqrt{25 - x^{2}}} = 0$

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 $a = 5$ and $b = x$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2}}{\sqrt{25 - x^{2}}} = 0$

Step 1.1.3

Simplify the denominator.

Tap for more steps...

Step 1.1.3.1

Rewrite $25$ as $5^{2}$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2}}{\sqrt{5^{2} - x^{2}}} = 0$

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 $a = 5$ and $b = x$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2}}{\sqrt{(5 + x) (5 - x)}} = 0$

Step 1.1.4

Multiply $\frac{x^{2}}{\sqrt{(5 + x) (5 - x)}}$ by $\frac{\sqrt{(5 + x) (5 - x)}}{\sqrt{(5 + x) (5 - x)}}$ .

$\sqrt{(5 + x) (5 - x)} - (\frac{x^{2}}{\sqrt{(5 + x) (5 - x)}} \cdot \frac{\sqrt{(5 + x) (5 - x)}}{\sqrt{(5 + x) (5 - x)}}) = 0$

Step 1.1.5

Combine and simplify the denominator.

Tap for more steps...

Step 1.1.5.1

Multiply $\frac{x^{2}}{\sqrt{(5 + x) (5 - x)}}$ by $\frac{\sqrt{(5 + x) (5 - x)}}{\sqrt{(5 + x) (5 - x)}}$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{\sqrt{(5 + x) (5 - x)} \sqrt{(5 + x) (5 - x)}} = 0$

Step 1.1.5.2

Raise $\sqrt{(5 + x) (5 - x)}$ to the power of $1$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1} \sqrt{(5 + x) (5 - x)}} = 0$

Step 1.1.5.3

Raise $\sqrt{(5 + x) (5 - x)}$ to the power of $1$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1} {\sqrt{(5 + x) (5 - x)}}^{1}} = 0$

Step 1.1.5.4

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

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{1 + 1}} = 0$

Step 1.1.5.5

Add $1$ and $1$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{\sqrt{(5 + x) (5 - x)}}^{2}} = 0$

Step 1.1.5.6

Rewrite ${\sqrt{(5 + x) (5 - x)}}^{2}$ as $(5 + x) (5 - x)$ .

Tap for more steps...

Step 1.1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + x) (5 - x)}$ as ${((5 + x) (5 - x))}^{\frac{1}{2}}$ .

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{({((5 + x) (5 - x))}^{\frac{1}{2}})}^{2}} = 0$

Step 1.1.5.6.2

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

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{\frac{1}{2} \cdot 2}} = 0$

Step 1.1.5.6.3

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

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{\frac{2}{2}}} = 0$

Step 1.1.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.5.6.4.1

Cancel the common factor.

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{\frac{2}{2}}} = 0$

Step 1.1.5.6.4.2

Rewrite the expression.

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{{((5 + x) (5 - x))}^{1}} = 0$

Step 1.1.5.6.5

Simplify.

$\sqrt{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{(5 + x) (5 - x)} = 0$

Step 1.2

To write $\sqrt{(5 + x) (5 - x)}$ as a fraction with a common denominator, multiply by $\frac{(5 + x) (5 - x)}{(5 + x) (5 - x)}$ .

$\sqrt{(5 + x) (5 - x)} \cdot \frac{(5 + x) (5 - x)}{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{(5 + x) (5 - x)} = 0$

Step 1.3

Combine $\sqrt{(5 + x) (5 - x)}$ and $\frac{(5 + x) (5 - x)}{(5 + x) (5 - x)}$ .

$\frac{\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x))}{(5 + x) (5 - x)} - \frac{x^{2} \sqrt{(5 + x) (5 - x)}}{(5 + x) (5 - x)} = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) - x^{2} \sqrt{(5 + x) (5 - x)}}{(5 + x) (5 - x)} = 0$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Factor $\sqrt{(5 + x) (5 - x)}$ out of $\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) - x^{2} \sqrt{(5 + x) (5 - x)}$ .

Tap for more steps...

Step 1.5.1.1

Factor $\sqrt{(5 + x) (5 - x)}$ out of $- x^{2} \sqrt{(5 + x) (5 - x)}$ .

$\frac{\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) + \sqrt{(5 + x) (5 - x)} (- x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.1.2

Factor $\sqrt{(5 + x) (5 - x)}$ out of $\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x)) + \sqrt{(5 + x) (5 - x)} (- x^{2})$ .

$\frac{\sqrt{(5 + x) (5 - x)} ((5 + x) (5 - x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.2

Expand $(5 + x) (5 - x)$ using the FOIL Method.

Tap for more steps...

Step 1.5.2.1

Apply the distributive property.

$\frac{\sqrt{(5 + x) (5 - x)} (5 (5 - x) + x (5 - x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.2.2

Apply the distributive property.

$\frac{\sqrt{(5 + x) (5 - x)} (5 \cdot 5 + 5 (- x) + x (5 - x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.2.3

Apply the distributive property.

$\frac{\sqrt{(5 + x) (5 - x)} (5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3

Simplify and combine like terms.

Tap for more steps...

Step 1.5.3.1

Simplify each term.

Tap for more steps...

Step 1.5.3.1.1

Multiply $5$ by $5$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 + 5 (- x) + x \cdot 5 + x (- x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.1.2

Multiply $- 1$ by $5$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 5 x + x \cdot 5 + x (- x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.1.3

Move $5$ to the left of $x$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 5 x + 5 \cdot x + x (- x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 5 x + 5 x - x \cdot x - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.1.5

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

Tap for more steps...

Step 1.5.3.1.5.1

Move $x$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 5 x + 5 x - (x \cdot x) - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.1.5.2

Multiply $x$ by $x$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 5 x + 5 x - x^{2} - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.2

Add $- 5 x$ and $5 x$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 + 0 - x^{2} - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.3.3

Add $25$ and $0$ .

$\frac{\sqrt{(5 + x) (5 - x)} (25 - x^{2} - x^{2})}{(5 + x) (5 - x)} = 0$

Step 1.5.4

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

$\frac{\sqrt{(5 + x) (5 - x)} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + x) (5 - x)}$ as ${((5 + x) (5 - x))}^{\frac{1}{2}}$ .

$\frac{{((5 + x) (5 - x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Expand $(5 + x) (5 - x)$ using the FOIL Method.

Tap for more steps...

Step 3.1.1

Apply the distributive property.

$\frac{{(5 (5 - x) + x (5 - x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.1.2

Apply the distributive property.

$\frac{{(5 \cdot 5 + 5 (- x) + x (5 - x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.1.3

Apply the distributive property.

$\frac{{(5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $5$ by $5$ .

$\frac{{(25 + 5 (- x) + x \cdot 5 + x (- x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.1.2

Multiply $- 1$ by $5$ .

$\frac{{(25 - 5 x + x \cdot 5 + x (- x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.1.3

Move $5$ to the left of $x$ .

$\frac{{(25 - 5 x + 5 x + x (- x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{{(25 - 5 x + 5 x - x \cdot x)}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.1.5

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

Tap for more steps...

Step 3.2.1.5.1

Move $x$ .

$\frac{{(25 - 5 x + 5 x - (x \cdot x))}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.1.5.2

Multiply $x$ by $x$ .

$\frac{{(25 - 5 x + 5 x - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.2

Add $- 5 x$ and $5 x$ .

$\frac{{(25 + 0 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 3.2.3

Add $25$ and $0$ .

$\frac{{(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$

Step 4

Find the LCD of the terms in the equation.

Tap for more steps...

Step 4.1

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

$(5 + x) (5 - x), 1$

Step 4.2

The LCM of one and any expression is the expression.

$(5 + x) (5 - x)$

Step 5

Multiply each term in $\frac{{(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$ by $(5 + x) (5 - x)$ to eliminate the fractions.

Tap for more steps...

Step 5.1

Multiply each term in $\frac{{(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} = 0$ by $(5 + x) (5 - x)$ .

$\frac{{(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} ((5 + x) (5 - x)) = 0 ((5 + x) (5 - x))$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $(5 + x) (5 - x)$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{{(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2})}{(5 + x) (5 - x)} ((5 + x) (5 - x)) = 0 ((5 + x) (5 - x))$

Step 5.2.1.2

Rewrite the expression.

${(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2}) = 0 ((5 + x) (5 - x))$

Step 5.2.2

Apply the distributive property.

${(25 - x^{2})}^{\frac{1}{2}} \cdot 25 + {(25 - x^{2})}^{\frac{1}{2}} (- 2 x^{2}) = 0 ((5 + x) (5 - x))$

Step 5.2.3

Reorder.

Tap for more steps...

Step 5.2.3.1

Move $25$ to the left of ${(25 - x^{2})}^{\frac{1}{2}}$ .

$25 \cdot {(25 - x^{2})}^{\frac{1}{2}} + {(25 - x^{2})}^{\frac{1}{2}} (- 2 x^{2}) = 0 ((5 + x) (5 - x))$

Step 5.2.3.2

Rewrite using the commutative property of multiplication.

$25 \cdot {(25 - x^{2})}^{\frac{1}{2}} - 2 {(25 - x^{2})}^{\frac{1}{2}} x^{2} = 0 ((5 + x) (5 - x))$

Step 5.2.3.3

Reorder factors in $25 {(25 - x^{2})}^{\frac{1}{2}} - 2 {(25 - x^{2})}^{\frac{1}{2}} x^{2}$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 ((5 + x) (5 - x))$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Expand $(5 + x) (5 - x)$ using the FOIL Method.

Tap for more steps...

Step 5.3.1.1

Apply the distributive property.

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (5 (5 - x) + x (5 - x))$

Step 5.3.1.2

Apply the distributive property.

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (5 \cdot 5 + 5 (- x) + x (5 - x))$

Step 5.3.1.3

Apply the distributive property.

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x))$

Step 5.3.2

Simplify and combine like terms.

Tap for more steps...

Step 5.3.2.1

Simplify each term.

Tap for more steps...

Step 5.3.2.1.1

Multiply $5$ by $5$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 + 5 (- x) + x \cdot 5 + x (- x))$

Step 5.3.2.1.2

Multiply $- 1$ by $5$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - 5 x + x \cdot 5 + x (- x))$

Step 5.3.2.1.3

Move $5$ to the left of $x$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - 5 x + 5 \cdot x + x (- x))$

Step 5.3.2.1.4

Rewrite using the commutative property of multiplication.

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - 5 x + 5 x - x \cdot x)$

Step 5.3.2.1.5

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

Tap for more steps...

Step 5.3.2.1.5.1

Move $x$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - 5 x + 5 x - (x \cdot x))$

Step 5.3.2.1.5.2

Multiply $x$ by $x$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - 5 x + 5 x - x^{2})$

Step 5.3.2.2

Add $- 5 x$ and $5 x$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 + 0 - x^{2})$

Step 5.3.2.3

Add $25$ and $0$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0 (25 - x^{2})$

Step 5.3.3

Multiply $0$ by $25 - x^{2}$ .

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 x^{2} {(25 - x^{2})}^{\frac{1}{2}} = 0$

Step 6

Solve the equation.

Tap for more steps...

Step 6.1

Factor the left side of the equation.

Tap for more steps...

Step 6.1.1

Add parentheses.

$25 {(25 - x^{2})}^{\frac{1}{2}} - 2 (x^{2} {(25 - x^{2})}^{\frac{1}{2}}) = 0$

Step 6.1.2

Let $u = {(25 - x^{2})}^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of ${(25 - x^{2})}^{\frac{1}{2}}$ .

$25 u - 2 x^{2} u = 0$

Step 6.1.3

Factor $u$ out of $25 u - 2 x^{2} u$ .

Tap for more steps...

Step 6.1.3.1

Factor $u$ out of $25 u$ .

$u \cdot 25 - 2 x^{2} u = 0$

Step 6.1.3.2

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

$u \cdot 25 + u (- 2 x^{2}) = 0$

Step 6.1.3.3

Factor $u$ out of $u \cdot 25 + u (- 2 x^{2})$ .

$u (25 - 2 x^{2}) = 0$

Step 6.1.4

Replace all occurrences of $u$ with ${(25 - x^{2})}^{\frac{1}{2}}$ .

${(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2}) = 0$

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 $0$ .

${(25 - x^{2})}^{\frac{1}{2}} = 0$

$25 - 2 x^{2} = 0$

Step 6.3

Set ${(25 - x^{2})}^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.3.1

Set ${(25 - x^{2})}^{\frac{1}{2}}$ equal to $0$ .

${(25 - x^{2})}^{\frac{1}{2}} = 0$

Step 6.3.2

Solve ${(25 - x^{2})}^{\frac{1}{2}} = 0$ for $x$ .

Tap for more steps...

Step 6.3.2.1

Set the $25 - x^{2}$ equal to $0$ .

$25 - x^{2} = 0$

Step 6.3.2.2

Solve for $x$ .

Tap for more steps...

Step 6.3.2.2.1

Subtract $25$ from both sides of the equation.

$- x^{2} = - 25$

Step 6.3.2.2.2

Divide each term in $- x^{2} = - 25$ by $- 1$ and simplify.

Tap for more steps...

Step 6.3.2.2.2.1

Divide each term in $- x^{2} = - 25$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 25}{- 1}$

Step 6.3.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.3.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 25}{- 1}$

Step 6.3.2.2.2.2.2

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

$x^{2} = \frac{- 25}{- 1}$

Step 6.3.2.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.3.2.2.2.3.1

Divide $- 25$ by $- 1$ .

$x^{2} = 25$

Step 6.3.2.2.3

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

$x = \pm \sqrt{25}$

Step 6.3.2.2.4

Simplify $\pm \sqrt{25}$ .

Tap for more steps...

Step 6.3.2.2.4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

Step 6.3.2.2.4.2

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

$x = \pm 5$

Step 6.3.2.2.5

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

Tap for more steps...

Step 6.3.2.2.5.1

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

$x = 5$

Step 6.3.2.2.5.2

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

$x = - 5$

Step 6.3.2.2.5.3

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

$x = 5, - 5$

Step 6.4

Set $25 - 2 x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.4.1

Set $25 - 2 x^{2}$ equal to $0$ .

$25 - 2 x^{2} = 0$

Step 6.4.2

Solve $25 - 2 x^{2} = 0$ for $x$ .

Tap for more steps...

Step 6.4.2.1

Subtract $25$ from both sides of the equation.

$- 2 x^{2} = - 25$

Step 6.4.2.2

Divide each term in $- 2 x^{2} = - 25$ by $- 2$ and simplify.

Tap for more steps...

Step 6.4.2.2.1

Divide each term in $- 2 x^{2} = - 25$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 25}{- 2}$

Step 6.4.2.2.2

Simplify the left side.

Tap for more steps...

Step 6.4.2.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 6.4.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 25}{- 2}$

Step 6.4.2.2.2.1.2

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

$x^{2} = \frac{- 25}{- 2}$

Step 6.4.2.2.3

Simplify the right side.

Tap for more steps...

Step 6.4.2.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{25}{2}$

Step 6.4.2.3

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

$x = \pm \sqrt{\frac{25}{2}}$

Step 6.4.2.4

Simplify $\pm \sqrt{\frac{25}{2}}$ .

Tap for more steps...

Step 6.4.2.4.1

Rewrite $\sqrt{\frac{25}{2}}$ as $\frac{\sqrt{25}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{25}}{\sqrt{2}}$

Step 6.4.2.4.2

Simplify the numerator.

Tap for more steps...

Step 6.4.2.4.2.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2}}}{\sqrt{2}}$

Step 6.4.2.4.2.2

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

$x = \pm \frac{5}{\sqrt{2}}$

Step 6.4.2.4.3

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{5}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.4.2.4.4

Combine and simplify the denominator.

Tap for more steps...

Step 6.4.2.4.4.1

Multiply $\frac{5}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{5 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 6.4.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.4.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.4.2.4.4.4

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

$x = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.4.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{5 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 6.4.2.4.4.6

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

Tap for more steps...

Step 6.4.2.4.4.6.1

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

$x = \pm \frac{5 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.4.2.4.4.6.2

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

$x = \pm \frac{5 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.4.2.4.4.6.3

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

$x = \pm \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.4.2.4.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.4.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{5 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 6.4.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{5 \sqrt{2}}{2^{1}}$

Step 6.4.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{5 \sqrt{2}}{2}$

Step 6.4.2.5

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

Tap for more steps...

Step 6.4.2.5.1

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

$x = \frac{5 \sqrt{2}}{2}$

Step 6.4.2.5.2

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

$x = - \frac{5 \sqrt{2}}{2}$

Step 6.4.2.5.3

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

$x = \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}$

Step 6.5

The final solution is all the values that make ${(25 - x^{2})}^{\frac{1}{2}} (25 - 2 x^{2}) = 0$ true.

$x = 5, - 5, \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}$

Step 7

Exclude the solutions that do not make $\sqrt{25 - x^{2}} - \frac{x^{2}}{\sqrt{25 - x^{2}}} = 0$ true.

$x = \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{2}}{2}, - \frac{5 \sqrt{2}}{2}$

Decimal Form:

$x = 3.53553390 \dots, - 3.53553390 \dots$