What is the general solution for 3tan^2(x)+5tan(x)-4=0 ?

The general solution for 3tan^2(x)+5tan(x)-4=0 is x=0.53352…+pin,x=-1.15377…+pin

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3tan^2(x)+5tan(x)-4=0

3 tan^{2} (x) + 5 tan (x) - 4 = 0

x = 0.53352 \dots + πn, x = - 1.15377 \dots + πn

Degrees

x = 30.56895 \dots^{\circ} + 18 0^{\circ} n, x = - 66.10663 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 tan^{2} (x) + 5 tan (x) - 4 = 0

Solve by substitution

3 tan^{2} (x) + 5 tan (x) - 4 = 0

Let:

tan (x) = u

3 u^{2} + 5 u - 4 = 0

3 u^{2} + 5 u - 4 = 0 : u = \frac{- 5 + 73}{6}, u = \frac{- 5 - 73}{6}

3 u^{2} + 5 u - 4 = 0

Solve with the quadratic formula

3 u^{2} + 5 u - 4 = 0

Quadratic Equation Formula:

For

a = 3, b = 5, c = - 4

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

5^{2} - 4 \cdot 3 (- 4) = 73

5^{2} - 4 \cdot 3 (- 4)

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 5^{2} + 48

5^{2} = 25

= 25 + 48

Add the numbers:

25 + 48 = 73

= 73

u_{1, 2} = \frac{- 5 \pm 73}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 5 + 73}{2 \cdot 3}, u_{2} = \frac{- 5 - 73}{2 \cdot 3}

u = \frac{- 5 + 73}{2 \cdot 3} : \frac{- 5 + 73}{6}

\frac{- 5 + 73}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 5 + 73}{6}

u = \frac{- 5 - 73}{2 \cdot 3} : \frac{- 5 - 73}{6}

\frac{- 5 - 73}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 5 - 73}{6}

The solutions to the quadratic equation are:

u = \frac{- 5 + 73}{6}, u = \frac{- 5 - 73}{6}

Substitute back

u = tan (x)

tan (x) = \frac{- 5 + 73}{6}, tan (x) = \frac{- 5 - 73}{6}

tan (x) = \frac{- 5 + 73}{6}, tan (x) = \frac{- 5 - 73}{6}

tan (x) = \frac{- 5 + 73}{6} : x = arctan (\frac{- 5 + 73}{6}) + πn

tan (x) = \frac{- 5 + 73}{6}

Apply trig inverse properties

tan (x) = \frac{- 5 + 73}{6}

General solutions for

tan (x) = \frac{- 5 + 73}{6}

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

x = arctan (\frac{- 5 + 73}{6}) + πn

x = arctan (\frac{- 5 + 73}{6}) + πn

tan (x) = \frac{- 5 - 73}{6} : x = arctan (\frac{- 5 - 73}{6}) + πn

tan (x) = \frac{- 5 - 73}{6}

Apply trig inverse properties

tan (x) = \frac{- 5 - 73}{6}

General solutions for

tan (x) = \frac{- 5 - 73}{6}

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

x = arctan (\frac{- 5 - 73}{6}) + πn

x = arctan (\frac{- 5 - 73}{6}) + πn

Combine all the solutions

x = arctan (\frac{- 5 + 73}{6}) + πn, x = arctan (\frac{- 5 - 73}{6}) + πn

Show solutions in decimal form

x = 0.53352 \dots + πn, x = - 1.15377 \dots + πn

cos (2 x) \cdot 2 = 0

csc^{3} (x) + csc^{2} (x) = 4 csc (x) + 4

2 sin^{2} (x) - 5 sin (x) - 3 = 0

6 + 6 sin (θ) = 4 cos^{2} (θ)

5 cos (x) + 5 sin (x) tan (x) = 10

What is the general solution for 3tan^2(x)+5tan(x)-4=0 ?
The general solution for 3tan^2(x)+5tan(x)-4=0 is x=0.53352…+pin,x=-1.15377…+pin