Algebra Examples

${(2 \sqrt{9 - x^{2}})}^{2}$

Step 1

Rewrite $9$ as $3^{2}$ .

${(2 \sqrt{3^{2} - x^{2}})}^{2}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

${(2 \sqrt{(3 + x) (3 - x)})}^{2}$

Step 3

Simplify by cancelling exponent with radical.

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Step 3.1

Apply the product rule to $2 \sqrt{(3 + x) (3 - x)}$ .

$2^{2} {\sqrt{(3 + x) (3 - x)}}^{2}$

Step 3.2

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

$4 {\sqrt{(3 + x) (3 - x)}}^{2}$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

$4 {((3 + x) (3 - x))}^{1}$

Step 3.3.5

Simplify.

$4 ((3 + x) (3 - x))$

Step 4

Expand $(3 + x) (3 - x)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$4 (3 (3 - x) + x (3 - x))$

Step 4.2

Apply the distributive property.

$4 (3 \cdot 3 + 3 (- x) + x (3 - x))$

Step 4.3

Apply the distributive property.

$4 (3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x))$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $3$ by $3$ .

$4 (9 + 3 (- x) + x \cdot 3 + x (- x))$

Step 5.1.2

Multiply $- 1$ by $3$ .

$4 (9 - 3 x + x \cdot 3 + x (- x))$

Step 5.1.3

Move $3$ to the left of $x$ .

$4 (9 - 3 x + 3 \cdot x + x (- x))$

Step 5.1.4

Rewrite using the commutative property of multiplication.

$4 (9 - 3 x + 3 x - x \cdot x)$

Step 5.1.5

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

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Step 5.1.5.1

Move $x$ .

$4 (9 - 3 x + 3 x - (x \cdot x))$

Step 5.1.5.2

Multiply $x$ by $x$ .

$4 (9 - 3 x + 3 x - x^{2})$

Step 5.2

Add $- 3 x$ and $3 x$ .

$4 (9 + 0 - x^{2})$

Step 5.3

Add $9$ and $0$ .

$4 (9 - x^{2})$

Step 6

Apply the distributive property.

$4 \cdot 9 + 4 (- x^{2})$

Step 7

Multiply.

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Step 7.1

Multiply $4$ by $9$ .

$36 + 4 (- x^{2})$

Step 7.2

Multiply $- 1$ by $4$ .

$36 - 4 x^{2}$