What is the general solution for 6=cot^2(x)+8cot(x) ?

The general solution for 6=cot^2(x)+8cot(x) is x=0.96653…+pin,x=3.02702…+pin

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6=cot^2(x)+8cot(x)

Solution

6 = cot^{2} (x) + 8 cot (x)

Solution

x = 0.96653 \dots + πn, x = 3.02702 \dots + πn

Degrees

x = 55.37818 \dots^{\circ} + 18 0^{\circ} n, x = 173.43588 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 = cot^{2} (x) + 8 cot (x)

Switch sides

cot^{2} (x) + 8 cot (x) = 6

Solve by substitution

cot^{2} (x) + 8 cot (x) = 6

Let:

cot (x) = u

u^{2} + 8 u = 6

u^{2} + 8 u = 6 : u = - 4 + 22, u = - 4 - 22

u^{2} + 8 u = 6

Move

6

to the left side

u^{2} + 8 u = 6

Subtract

6

from both sides

u^{2} + 8 u - 6 = 6 - 6

Simplify

u^{2} + 8 u - 6 = 0

u^{2} + 8 u - 6 = 0

Solve with the quadratic formula

u^{2} + 8 u - 6 = 0

Quadratic Equation Formula:

For

a = 1, b = 8, c = - 6

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

8^{2} - 4 \cdot 1 \cdot (- 6) = 222

8^{2} - 4 \cdot 1 \cdot (- 6)

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 8^{2} + 24

8^{2} = 64

= 64 + 24

Add the numbers:

64 + 24 = 88

= 88

Prime factorization of

88 : 2^{3} \cdot 11

88

88

divides by

288 = 44 \cdot 2

= 2 \cdot 44

44

divides by

244 = 22 \cdot 2

= 2 \cdot 2 \cdot 22

22

divides by

222 = 11 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 11

2, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 11

= 2^{3} \cdot 11

= 2^{3} \cdot 11

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 11

Apply radical rule:

n ab = n a n b

= 2^{2} 2 \cdot 11

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 11

Refine

= 222

u_{1, 2} = \frac{- 8 \pm 2 22}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 8 + 2 22}{2 \cdot 1}, u_{2} = \frac{- 8 - 2 22}{2 \cdot 1}

u = \frac{- 8 + 2 22}{2 \cdot 1} : - 4 + 22

\frac{- 8 + 2 22}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8 + 2 22}{2}

Factor

- 8 + 222 : 2 (- 4 + 22)

- 8 + 222

Rewrite as

= - 2 \cdot 4 + 222

Factor out common term

2

= 2 (- 4 + 22)

= \frac{2 ( - 4 + 22 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 4 + 22

u = \frac{- 8 - 2 22}{2 \cdot 1} : - 4 - 22

\frac{- 8 - 2 22}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8 - 2 22}{2}

Factor

- 8 - 222 : - 2 (4 + 22)

- 8 - 222

Rewrite as

= - 2 \cdot 4 - 222

Factor out common term

2

= - 2 (4 + 22)

= - \frac{2 ( 4 + 22 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (4 + 22)

Negate

- (4 + 22) = - 4 - 22

= - 4 - 22

The solutions to the quadratic equation are:

u = - 4 + 22, u = - 4 - 22

Substitute back

u = cot (x)

cot (x) = - 4 + 22, cot (x) = - 4 - 22

cot (x) = - 4 + 22, cot (x) = - 4 - 22

cot (x) = - 4 + 22 : x = arccot (- 4 + 22) + πn

cot (x) = - 4 + 22

Apply trig inverse properties

cot (x) = - 4 + 22

General solutions for

cot (x) = - 4 + 22

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (- 4 + 22) + πn

x = arccot (- 4 + 22) + πn

cot (x) = - 4 - 22 : x = arccot (- 4 - 22) + πn

cot (x) = - 4 - 22

Apply trig inverse properties

cot (x) = - 4 - 22

General solutions for

cot (x) = - 4 - 22

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 4 - 22) + πn

x = arccot (- 4 - 22) + πn

Combine all the solutions

x = arccot (- 4 + 22) + πn, x = arccot (- 4 - 22) + πn

Show solutions in decimal form

x = 0.96653 \dots + πn, x = 3.02702 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6=cot^2(x)+8cot(x) ?
The general solution for 6=cot^2(x)+8cot(x) is x=0.96653…+pin,x=3.02702…+pin