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

The general solution for 3tan^2(x)+4tan(x)-3=0 is x=0.49139…+pin,x=-1.07939…+pin

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3tan^2(x)+4tan(x)-3=0

Solution

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

Solution

x = 0.49139 \dots + πn, x = - 1.07939 \dots + πn

Degrees

x = 28.15496 \dots^{\circ} + 18 0^{\circ} n, x = - 61.84503 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

3 u^{2} + 4 u - 3 = 0 : u = \frac{- 2 + 13}{3}, u = - \frac{2 + 13}{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}

4^{2} - 4 \cdot 3 (- 3) = 213

4^{2} - 4 \cdot 3 (- 3)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 3 \cdot 3

Multiply the numbers:

4 \cdot 3 \cdot 3 = 36

= 4^{2} + 36

4^{2} = 16

= 16 + 36

Add the numbers:

16 + 36 = 52

= 52

Prime factorization of

52 : 2^{2} \cdot 13

52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

Apply radical rule:

= 13 2^{2}

Apply radical rule:

2^{2} = 2

= 213

u_{1, 2} = \frac{- 4 \pm 2 13}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 4 + 2 13}{2 \cdot 3}, u_{2} = \frac{- 4 - 2 13}{2 \cdot 3}

u = \frac{- 4 + 2 13}{2 \cdot 3} : \frac{- 2 + 13}{3}

\frac{- 4 + 2 13}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4 + 2 13}{6}

Factor

- 4 + 213 : 2 (- 2 + 13)

- 4 + 213

Rewrite as

= - 2 \cdot 2 + 213

Factor out common term

2

= 2 (- 2 + 13)

= \frac{2 ( - 2 + 13 )}{6}

Cancel the common factor:

2

= \frac{- 2 + 13}{3}

u = \frac{- 4 - 2 13}{2 \cdot 3} : - \frac{2 + 13}{3}

\frac{- 4 - 2 13}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4 - 2 13}{6}

Factor

- 4 - 213 : - 2 (2 + 13)

- 4 - 213

Rewrite as

= - 2 \cdot 2 - 213

Factor out common term

2

= - 2 (2 + 13)

= - \frac{2 ( 2 + 13 )}{6}

Cancel the common factor:

2

= - \frac{2 + 13}{3}

The solutions to the quadratic equation are:

u = \frac{- 2 + 13}{3}, u = - \frac{2 + 13}{3}

Substitute back

u = tan (x)

tan (x) = \frac{- 2 + 13}{3}, tan (x) = - \frac{2 + 13}{3}

tan (x) = \frac{- 2 + 13}{3}, tan (x) = - \frac{2 + 13}{3}

tan (x) = \frac{- 2 + 13}{3} : x = arctan (\frac{- 2 + 13}{3}) + πn

tan (x) = \frac{- 2 + 13}{3}

Apply trig inverse properties

tan (x) = \frac{- 2 + 13}{3}

General solutions for

tan (x) = \frac{- 2 + 13}{3}

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

x = arctan (\frac{- 2 + 13}{3}) + πn

x = arctan (\frac{- 2 + 13}{3}) + πn

tan (x) = - \frac{2 + 13}{3} : x = arctan (- \frac{2 + 13}{3}) + πn

tan (x) = - \frac{2 + 13}{3}

Apply trig inverse properties

tan (x) = - \frac{2 + 13}{3}

General solutions for

tan (x) = - \frac{2 + 13}{3}

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

x = arctan (- \frac{2 + 13}{3}) + πn

x = arctan (- \frac{2 + 13}{3}) + πn

Combine all the solutions

x = arctan (\frac{- 2 + 13}{3}) + πn, x = arctan (- \frac{2 + 13}{3}) + πn

Show solutions in decimal form

x = 0.49139 \dots + πn, x = - 1.07939 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan^2(x)+4tan(x)-3=0 ?
The general solution for 3tan^2(x)+4tan(x)-3=0 is x=0.49139…+pin,x=-1.07939…+pin