Trigonometry Examples

$\frac{7 (10 + c) (10 - c)}{10 + c} + \frac{10 (10 + c) (10 - c)}{10 - c} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1

Factor each term.

Tap for more steps...

Step 1.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.1.1

Reduce the expression $\frac{7 (10 + c) (10 - c)}{10 + c}$ by cancelling the common factors.

Tap for more steps...

Step 1.1.1.1

Cancel the common factor.

$\frac{7 (10 + c) (10 - c)}{10 + c} + \frac{10 (10 + c) (10 - c)}{10 - c} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.1.1.2

Rewrite the expression.

$\frac{7 (10 - c)}{1} + \frac{10 (10 + c) (10 - c)}{10 - c} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.1.2

Divide $7 (10 - c)$ by $1$ .

$7 (10 - c) + \frac{10 (10 + c) (10 - c)}{10 - c} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.2.1

Reduce the expression $\frac{10 (10 + c) (10 - c)}{10 - c}$ by cancelling the common factors.

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$7 (10 - c) + \frac{10 (10 + c) (10 - c)}{10 - c} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.2.1.2

Rewrite the expression.

$7 (10 - c) + \frac{10 (10 + c)}{1} = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.2.2

Divide $10 (10 + c)$ by $1$ .

$7 (10 - c) + 10 (10 + c) = \frac{10 (10 + c) (10 - c)}{1}$

Step 1.3

Divide $10 (10 + c) (10 - c)$ by $1$ .

$7 (10 - c) + 10 (10 + c) = 10 (10 + c) (10 - c)$

Step 2

Solve the equation.

Tap for more steps...

Step 2.1

Simplify $7 (10 - c) + 10 (10 + c)$ .

Tap for more steps...

Step 2.1.1

Simplify each term.

Tap for more steps...

Step 2.1.1.1

Apply the distributive property.

$7 \cdot 10 + 7 (- c) + 10 (10 + c) = 10 (10 + c) (10 - c)$

Step 2.1.1.2

Multiply $7$ by $10$ .

$70 + 7 (- c) + 10 (10 + c) = 10 (10 + c) (10 - c)$

Step 2.1.1.3

Multiply $- 1$ by $7$ .

$70 - 7 c + 10 (10 + c) = 10 (10 + c) (10 - c)$

Step 2.1.1.4

Apply the distributive property.

$70 - 7 c + 10 \cdot 10 + 10 c = 10 (10 + c) (10 - c)$

Step 2.1.1.5

Multiply $10$ by $10$ .

$70 - 7 c + 100 + 10 c = 10 (10 + c) (10 - c)$

Step 2.1.2

Simplify by adding terms.

Tap for more steps...

Step 2.1.2.1

Add $70$ and $100$ .

$- 7 c + 170 + 10 c = 10 (10 + c) (10 - c)$

Step 2.1.2.2

Add $- 7 c$ and $10 c$ .

$3 c + 170 = 10 (10 + c) (10 - c)$

Step 2.2

Simplify $10 (10 + c) (10 - c)$ .

Tap for more steps...

Step 2.2.1

Simplify by multiplying through.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$3 c + 170 = (10 \cdot 10 + 10 c) (10 - c)$

Step 2.2.1.2

Multiply $10$ by $10$ .

$3 c + 170 = (100 + 10 c) (10 - c)$

Step 2.2.2

Expand $(100 + 10 c) (10 - c)$ using the FOIL Method.

Tap for more steps...

Step 2.2.2.1

Apply the distributive property.

$3 c + 170 = 100 (10 - c) + 10 c (10 - c)$

Step 2.2.2.2

Apply the distributive property.

$3 c + 170 = 100 \cdot 10 + 100 (- c) + 10 c (10 - c)$

Step 2.2.2.3

Apply the distributive property.

$3 c + 170 = 100 \cdot 10 + 100 (- c) + 10 c \cdot 10 + 10 c (- c)$

Step 2.2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.2.3.1

Simplify each term.

Tap for more steps...

Step 2.2.3.1.1

Multiply $100$ by $10$ .

$3 c + 170 = 1000 + 100 (- c) + 10 c \cdot 10 + 10 c (- c)$

Step 2.2.3.1.2

Multiply $- 1$ by $100$ .

$3 c + 170 = 1000 - 100 c + 10 c \cdot 10 + 10 c (- c)$

Step 2.2.3.1.3

Multiply $10$ by $10$ .

$3 c + 170 = 1000 - 100 c + 100 c + 10 c (- c)$

Step 2.2.3.1.4

Rewrite using the commutative property of multiplication.

$3 c + 170 = 1000 - 100 c + 100 c + 10 \cdot - 1 c \cdot c$

Step 2.2.3.1.5

Multiply $c$ by $c$ by adding the exponents.

Tap for more steps...

Step 2.2.3.1.5.1

Move $c$ .

$3 c + 170 = 1000 - 100 c + 100 c + 10 \cdot - 1 (c \cdot c)$

Step 2.2.3.1.5.2

Multiply $c$ by $c$ .

$3 c + 170 = 1000 - 100 c + 100 c + 10 \cdot - 1 c^{2}$

Step 2.2.3.1.6

Multiply $10$ by $- 1$ .

$3 c + 170 = 1000 - 100 c + 100 c - 10 c^{2}$

Step 2.2.3.2

Add $- 100 c$ and $100 c$ .

$3 c + 170 = 1000 + 0 - 10 c^{2}$

Step 2.2.3.3

Add $1000$ and $0$ .

$3 c + 170 = 1000 - 10 c^{2}$

Step 2.3

Add $10 c^{2}$ to both sides of the equation.

$3 c + 170 + 10 c^{2} = 1000$

Step 2.4

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 2.4.1

Subtract $1000$ from both sides of the equation.

$3 c + 170 + 10 c^{2} - 1000 = 0$

Step 2.4.2

Subtract $1000$ from $170$ .

$3 c + 10 c^{2} - 830 = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 10$ , $b = 3$ , and $c = - 830$ into the quadratic formula and solve for $c$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (10 \cdot - 830)}}{2 \cdot 10}$

Step 2.7

Simplify.

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

Step 2.7.1.1

Raise $3$ to the power of $2$ .

$c = \frac{- 3 \pm \sqrt{9 - 4 \cdot 10 \cdot - 830}}{2 \cdot 10}$

Step 2.7.1.2

Multiply $- 4 \cdot 10 \cdot - 830$ .

Tap for more steps...

Step 2.7.1.2.1

Multiply $- 4$ by $10$ .

$c = \frac{- 3 \pm \sqrt{9 - 40 \cdot - 830}}{2 \cdot 10}$

Step 2.7.1.2.2

Multiply $- 40$ by $- 830$ .

$c = \frac{- 3 \pm \sqrt{9 + 33200}}{2 \cdot 10}$

Step 2.7.1.3

Add $9$ and $33200$ .

$c = \frac{- 3 \pm \sqrt{33209}}{2 \cdot 10}$

Step 2.7.2

Multiply $2$ by $10$ .

$c = \frac{- 3 \pm \sqrt{33209}}{20}$

Step 2.8

The final answer is the combination of both solutions.

$c = - \frac{3 - \sqrt{33209}}{20}, - \frac{3 + \sqrt{33209}}{20}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$c = - \frac{3 - \sqrt{33209}}{20}, - \frac{3 + \sqrt{33209}}{20}$

Decimal Form:

$c = 8.96166834 \dots, - 9.26166834 \dots$