Trigonometry Examples

$10 b - b^{2} = - 5 \sqrt{25 - b^{2}} + {(\sqrt{25 - b^{2}})}^{2}$

Step 1

Simplify each term.

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Step 1.1

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

$10 b - b^{2} = - 5 \sqrt{5^{2} - b^{2}} + {(\sqrt{25 - b^{2}})}^{2}$

Step 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 $b = b$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {(\sqrt{25 - b^{2}})}^{2}$

Step 1.3

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

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {\sqrt{5^{2} - b^{2}}}^{2}$

Step 1.4

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 $b = b$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {\sqrt{(5 + b) (5 - b)}}^{2}$

Step 1.5

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

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Step 1.5.1

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

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {({((5 + b) (5 - b))}^{\frac{1}{2}})}^{2}$

Step 1.5.2

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

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {((5 + b) (5 - b))}^{\frac{1}{2} \cdot 2}$

Step 1.5.3

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

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {((5 + b) (5 - b))}^{\frac{2}{2}}$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {((5 + b) (5 - b))}^{\frac{2}{2}}$

Step 1.5.4.2

Rewrite the expression.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + {((5 + b) (5 - b))}^{1}$

Step 1.5.5

Simplify.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + (5 + b) (5 - b)$

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 5 (5 - b) + b (5 - b)$

Step 1.6.2

Apply the distributive property.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 5 \cdot 5 + 5 (- b) + b (5 - b)$

Step 1.6.3

Apply the distributive property.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 5 \cdot 5 + 5 (- b) + b \cdot 5 + b (- b)$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $5$ by $5$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 + 5 (- b) + b \cdot 5 + b (- b)$

Step 1.7.1.2

Multiply $- 1$ by $5$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - 5 b + b \cdot 5 + b (- b)$

Step 1.7.1.3

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

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - 5 b + 5 \cdot b + b (- b)$

Step 1.7.1.4

Rewrite using the commutative property of multiplication.

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - 5 b + 5 b - b \cdot b$

Step 1.7.1.5

Multiply $b$ by $b$ by adding the exponents.

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Step 1.7.1.5.1

Move $b$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - 5 b + 5 b - (b \cdot b)$

Step 1.7.1.5.2

Multiply $b$ by $b$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - 5 b + 5 b - b^{2}$

Step 1.7.2

Add $- 5 b$ and $5 b$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 + 0 - b^{2}$

Step 1.7.3

Add $25$ and $0$ .

$10 b - b^{2} = - 5 \sqrt{(5 + b) (5 - b)} + 25 - b^{2}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$b = 0$

Step 3