What is the general solution for 3cot^2(x)-4cot(x)+sqrt(3)=0 ?

The general solution for 3cot^2(x)-4cot(x)+sqrt(3)=0 is No Solution for x\in\mathbb{R}

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3cot^2(x)-4cot(x)+sqrt(3)=0

Solution

3 cot^{2} (x) - 4 cot (x) + 3 = 0

Solution

No Solution for x \in R

Solution steps

3 cot^{2} (x) - 4 cot (x) + 3 = 0

Solve by substitution

3 cot^{2} (x) - 4 cot (x) + 3 = 0

Let:

cot (x) = u

3 u^{2} - 4 u + 3 = 0

3 u^{2} - 4 u + 3 = 0 : u = \frac{2}{3} + i \frac{- 4 + 3 3}{3}, u = \frac{2}{3} - i \frac{- 4 + 3 3}{3}

3 u^{2} - 4 u + 3 = 0

Solve with the quadratic formula

3 u^{2} - 4 u + 3 = 0

Quadratic Equation Formula:

For

a = 3, b = - 4, c = 3

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 3 3}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 3 3}{2 \cdot 3}

Simplify

(- 4)^{2} - 4 \cdot 33 : 2 i - 4 + 33

(- 4)^{2} - 4 \cdot 33

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 4)^{2} = 4^{2}

= 4^{2} - 4 \cdot 33

Multiply the numbers:

4 \cdot 3 = 12

= 4^{2} - 123

Apply imaginary number rule:

- a = i a

= i 123 - 4^{2}

- 4^{2} + 123 = 2 - 4 + 33

- 4^{2} + 123

4^{2} = 16

= - 16 + 123

Factor

- 16 + 123 : 4 (- 4 + 33)

- 16 + 123

Rewrite as

= - 4 \cdot 4 + 4 \cdot 33

Factor out common term

4

= 4 (- 4 + 33)

= 4 (- 4 + 33)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 4 33 - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= 2 33 - 4

= 2 i 33 - 4

u_{1, 2} = \frac{- ( - 4 ) \pm 2 i - 4 + 3 3}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 2 i - 4 + 3 3}{2 \cdot 3}, u_{2} = \frac{- ( - 4 ) - 2 i - 4 + 3 3}{2 \cdot 3}

u = \frac{- ( - 4 ) + 2 i - 4 + 3 3}{2 \cdot 3} : \frac{2}{3} + i \frac{- 4 + 3 3}{3}

\frac{- ( - 4 ) + 2 i - 4 + 3 3}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{4 + 2 i - 4 + 3 3}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4 + 2 i 3 3 - 4}{6}

Factor

4 + 2 i - 4 + 33 : 2 (2 + i - 4 + 33)

4 + 2 i - 4 + 33

Rewrite as

= 2 \cdot 2 + 2 i - 4 + 33

Factor out common term

2

= 2 (2 + i - 4 + 33)

= \frac{2 ( 2 + i - 4 + 3 3 )}{6}

Cancel the common factor:

2

= \frac{2 + i 3 3 - 4}{3}

Rewrite

\frac{2 + i 3 3 - 4}{3}

in standard complex form:

\frac{2}{3} + \frac{3 3 - 4}{3} i

\frac{2 + i 3 3 - 4}{3}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 + i 3 3 - 4}{3} = \frac{2}{3} + \frac{i 3 3 - 4}{3}

= \frac{2}{3} + \frac{i 3 3 - 4}{3}

= \frac{2}{3} + \frac{3 3 - 4}{3} i

u = \frac{- ( - 4 ) - 2 i - 4 + 3 3}{2 \cdot 3} : \frac{2}{3} - i \frac{- 4 + 3 3}{3}

\frac{- ( - 4 ) - 2 i - 4 + 3 3}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{4 - 2 i - 4 + 3 3}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4 - 2 i 3 3 - 4}{6}

Factor

4 - 2 i - 4 + 33 : 2 (2 - i - 4 + 33)

4 - 2 i - 4 + 33

Rewrite as

= 2 \cdot 2 - 2 i - 4 + 33

Factor out common term

2

= 2 (2 - i - 4 + 33)

= \frac{2 ( 2 - i - 4 + 3 3 )}{6}

Cancel the common factor:

2

= \frac{2 - i 3 3 - 4}{3}

Rewrite

\frac{2 - i 3 3 - 4}{3}

in standard complex form:

\frac{2}{3} - \frac{3 3 - 4}{3} i

\frac{2 - i 3 3 - 4}{3}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 - i 3 3 - 4}{3} = \frac{2}{3} - \frac{i 3 3 - 4}{3}

= \frac{2}{3} - \frac{i 3 3 - 4}{3}

= \frac{2}{3} - \frac{3 3 - 4}{3} i

The solutions to the quadratic equation are:

u = \frac{2}{3} + i \frac{- 4 + 3 3}{3}, u = \frac{2}{3} - i \frac{- 4 + 3 3}{3}

Substitute back

u = cot (x)

cot (x) = \frac{2}{3} + i \frac{- 4 + 3 3}{3}, cot (x) = \frac{2}{3} - i \frac{- 4 + 3 3}{3}

cot (x) = \frac{2}{3} + i \frac{- 4 + 3 3}{3}, cot (x) = \frac{2}{3} - i \frac{- 4 + 3 3}{3}

cot (x) = \frac{2}{3} + i \frac{- 4 + 3 3}{3} :

No Solution

cot (x) = \frac{2}{3} + i \frac{- 4 + 3 3}{3}

No Solution

cot (x) = \frac{2}{3} - i \frac{- 4 + 3 3}{3} :

No Solution

cot (x) = \frac{2}{3} - i \frac{- 4 + 3 3}{3}

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(x)-4cot(x)+sqrt(3)=0 ?
The general solution for 3cot^2(x)-4cot(x)+sqrt(3)=0 is No Solution for x\in\mathbb{R}