Trigonometry Examples

$r^{2} - 6 r - 83 = - 10$

Step 1

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

Tap for more steps...

Step 1.1

Add $10$ to both sides of the equation.

$r^{2} - 6 r - 83 + 10 = 0$

Step 1.2

Add $- 83$ and $10$ .

$r^{2} - 6 r - 73 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 73$ into the quadratic formula and solve for $r$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 73)}}{2 \cdot 1}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Raise $- 6$ to the power of $2$ .

$r = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 73}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 73$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $1$ .

$r = \frac{6 \pm \sqrt{36 - 4 \cdot - 73}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 73$ .

$r = \frac{6 \pm \sqrt{36 + 292}}{2 \cdot 1}$

Step 4.1.3

Add $36$ and $292$ .

$r = \frac{6 \pm \sqrt{328}}{2 \cdot 1}$

Step 4.1.4

Rewrite $328$ as $2^{2} \cdot 82$ .

Tap for more steps...

Step 4.1.4.1

Factor $4$ out of $328$ .

$r = \frac{6 \pm \sqrt{4 (82)}}{2 \cdot 1}$

Step 4.1.4.2

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

$r = \frac{6 \pm \sqrt{2^{2} \cdot 82}}{2 \cdot 1}$

Step 4.1.5

Pull terms out from under the radical.

$r = \frac{6 \pm 2 \sqrt{82}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$r = \frac{6 \pm 2 \sqrt{82}}{2}$

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{82}}{2}$ .

$r = 3 \pm \sqrt{82}$

Step 5

The final answer is the combination of both solutions.

$r = 3 + \sqrt{82}, 3 - \sqrt{82}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$r = 3 + \sqrt{82}, 3 - \sqrt{82}$

Decimal Form:

$r = 12.05538513 \dots, - 6.05538513 \dots$