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

The general solution for 3tan^2(x)+5tan(x)+2=0 is x=-0.58800…+pin,x=(3pi)/4+pin

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

Solution

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

Solution

x = - 0.58800 \dots + πn, x = \frac{3 π}{4} + πn

Degrees

x = - 33.69006 \dots^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

3 u^{2} + 5 u + 2 = 0 : u = - \frac{2}{3}, u = - 1

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 3, b = 5, c = 2

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

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

5^{2} - 4 \cdot 3 \cdot 2 = 1

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

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} - 24

5^{2} = 25

= 25 - 24

Subtract the numbers:

25 - 24 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

u = \frac{- 5 + 1}{2 \cdot 3} : - \frac{2}{3}

\frac{- 5 + 1}{2 \cdot 3}

Add/Subtract the numbers:

- 5 + 1 = - 4

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{6}

Apply the fraction rule:

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

= - \frac{4}{6}

Cancel the common factor:

2

= - \frac{2}{3}

u = \frac{- 5 - 1}{2 \cdot 3} : - 1

\frac{- 5 - 1}{2 \cdot 3}

Subtract the numbers:

- 5 - 1 = - 6

= \frac{- 6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 6}{6}

Apply the fraction rule:

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

= - \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = - \frac{2}{3}, u = - 1

Substitute back

u = tan (x)

tan (x) = - \frac{2}{3}, tan (x) = - 1

tan (x) = - \frac{2}{3}, tan (x) = - 1

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{2}{3}) + πn

x = arctan (- \frac{2}{3}) + πn

tan (x) = - 1 : x = \frac{3 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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = arctan (- \frac{2}{3}) + πn, x = \frac{3 π}{4} + πn

Show solutions in decimal form

x = - 0.58800 \dots + πn, x = \frac{3 π}{4} + πn

Graph

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan^2(x)+5tan(x)+2=0 ?
The general solution for 3tan^2(x)+5tan(x)+2=0 is x=-0.58800…+pin,x=(3pi)/4+pin