What is the general solution for 4tan(x)-tan^2(x)=1 ?

The general solution for 4tan(x)-tan^2(x)=1 is x=0.26179…+pin,x=1.30899…+pin

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4tan(x)-tan^2(x)=1

Solution

4 tan (x) - tan^{2} (x) = 1

Solution

x = 0.26179 \dots + πn, x = 1.30899 \dots + πn

Degrees

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

Solution steps

4 tan (x) - tan^{2} (x) = 1

Solve by substitution

4 tan (x) - tan^{2} (x) = 1

Let:

tan (x) = u

4 u - u^{2} = 1

4 u - u^{2} = 1 : u = 2 - 3, u = 2 + 3

4 u - u^{2} = 1

Move

1

to the left side

4 u - u^{2} = 1

Subtract

1

from both sides

4 u - u^{2} - 1 = 1 - 1

Simplify

4 u - u^{2} - 1 = 0

4 u - u^{2} - 1 = 0

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} + 4 u - 1 = 0

Solve with the quadratic formula

- u^{2} + 4 u - 1 = 0

Quadratic Equation Formula:

For

a = - 1, b = 4, c = - 1

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

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

4^{2} - 4 (- 1) (- 1) = 23

4^{2} - 4 (- 1) (- 1)

Apply rule

- (- a) = a

= 4^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4^{2} - 4

4^{2} = 16

= 16 - 4

Subtract the numbers:

16 - 4 = 12

= 12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

n ab = n a n b

= 3 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 23

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

Separate the solutions

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

u = \frac{- 4 + 2 3}{2 ( - 1 )} : 2 - 3

\frac{- 4 + 2 3}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 2 3}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 4 + 23 = - (4 - 23)

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

Factor

4 - 23 : 2 (2 - 3)

4 - 23

Rewrite as

= 2 \cdot 2 - 23

Factor out common term

2

= 2 (2 - 3)

= \frac{2 ( 2 - 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 - 3

u = \frac{- 4 - 2 3}{2 ( - 1 )} : 2 + 3

\frac{- 4 - 2 3}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 2 3}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

- 4 - 23 = - (4 + 23)

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

Factor

4 + 23 : 2 (2 + 3)

4 + 23

Rewrite as

= 2 \cdot 2 + 23

Factor out common term

2

= 2 (2 + 3)

= \frac{2 ( 2 + 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 2 + 3

The solutions to the quadratic equation are:

u = 2 - 3, u = 2 + 3

Substitute back

u = tan (x)

tan (x) = 2 - 3, tan (x) = 2 + 3

tan (x) = 2 - 3, tan (x) = 2 + 3

tan (x) = 2 - 3 : x = arctan (2 - 3) + πn

tan (x) = 2 - 3

Apply trig inverse properties

tan (x) = 2 - 3

General solutions for

tan (x) = 2 - 3

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2 - 3) + πn

x = arctan (2 - 3) + πn

tan (x) = 2 + 3 : x = arctan (2 + 3) + πn

tan (x) = 2 + 3

Apply trig inverse properties

tan (x) = 2 + 3

General solutions for

tan (x) = 2 + 3

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2 + 3) + πn

x = arctan (2 + 3) + πn

Combine all the solutions

x = arctan (2 - 3) + πn, x = arctan (2 + 3) + πn

Show solutions in decimal form

x = 0.26179 \dots + πn, x = 1.30899 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan(x)-tan^2(x)=1 ?
The general solution for 4tan(x)-tan^2(x)=1 is x=0.26179…+pin,x=1.30899…+pin