What is the general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 ?

The general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 is x= pi/4+pin,x=(5pi)/6+pin,x= pi/6+pin

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3tan^3(x)-3tan^2(x)-tan(x)+1=0

Solution

3 tan^{3} (x) - 3 tan^{2} (x) - tan (x) + 1 = 0

Solution

x = \frac{π}{4} + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{3} (x) - 3 tan^{2} (x) - tan (x) + 1 = 0

Solve by substitution

3 tan^{3} (x) - 3 tan^{2} (x) - tan (x) + 1 = 0

Let:

tan (x) = u

3 u^{3} - 3 u^{2} - u + 1 = 0

3 u^{3} - 3 u^{2} - u + 1 = 0 : u = 1, u = - \frac{3}{3}, u = \frac{3}{3}

3 u^{3} - 3 u^{2} - u + 1 = 0

Factor

3 u^{3} - 3 u^{2} - u + 1 : (u - 1) (3 u + 1) (3 u - 1)

3 u^{3} - 3 u^{2} - u + 1

= (3 u^{3} - 3 u^{2}) + (- u + 1)

Factor out

- 1

from

- u + 1 : - (u - 1)

- u + 1

Factor out common term

- 1

= - (u - 1)

Factor out

3 u^{2}

from

3 u^{3} - 3 u^{2} : 3 u^{2} (u - 1)

3 u^{3} - 3 u^{2}

Apply exponent rule:

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

u^{3} = u u^{2}

= 3 u u^{2} - 3 u^{2}

Factor out common term

3 u^{2}

= 3 u^{2} (u - 1)

= - (u - 1) + 3 u^{2} (u - 1)

Factor out common term

u - 1

= (u - 1) (3 u^{2} - 1)

Factor

3 u^{2} - 1 : (3 u + 1) (3 u - 1)

3 u^{2} - 1

Rewrite

3 u^{2} - 1

(3 u)^{2} - 1^{2}

3 u^{2} - 1

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} u^{2} - 1

Rewrite

1

1^{2}

= (3)^{2} u^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(3)^{2} u^{2} = (3 u)^{2}

= (3 u)^{2} - 1^{2}

= (3 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(3 u)^{2} - 1^{2} = (3 u + 1) (3 u - 1)

= (3 u + 1) (3 u - 1)

= (u - 1) (3 u + 1) (3 u - 1)

(u - 1) (3 u + 1) (3 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 1 = 0 or 3 u + 1 = 0 or 3 u - 1 = 0

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

3 u + 1 = 0 : u = - \frac{3}{3}

3 u + 1 = 0

Move

1

to the right side

3 u + 1 = 0

Subtract

1

from both sides

3 u + 1 - 1 = 0 - 1

Simplify

3 u = - 1

3 u = - 1

Divide both sides by

3

3 u = - 1

Divide both sides by

3

\frac{3 u}{3} = \frac{- 1}{3}

Simplify

\frac{3 u}{3} = \frac{- 1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{3}

Rationalize

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

Solve

3 u - 1 = 0 : u = \frac{3}{3}

3 u - 1 = 0

Move

1

to the right side

3 u - 1 = 0

Add

1

to both sides

3 u - 1 + 1 = 0 + 1

Simplify

3 u = 1

3 u = 1

Divide both sides by

3

3 u = 1

Divide both sides by

3

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} = \frac{1}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

The solutions are

u = 1, u = - \frac{3}{3}, u = \frac{3}{3}

Substitute back

u = tan (x)

tan (x) = 1, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 1, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

tan (x) = - \frac{3}{3} : x = \frac{5 π}{6} + πn

tan (x) = - \frac{3}{3}

General solutions for

tan (x) = - \frac{3}{3}

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

tan (x) = \frac{3}{3} : x = \frac{π}{6} + πn

tan (x) = \frac{3}{3}

General solutions for

tan (x) = \frac{3}{3}

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 ?
The general solution for 3tan^3(x)-3tan^2(x)-tan(x)+1=0 is x= pi/4+pin,x=(5pi)/6+pin,x= pi/6+pin