What is the general solution for 2tan^2(x)+9tan(x)+3=0 ?

The general solution for 2tan^2(x)+9tan(x)+3=0 is x=-0.34780…+pin,x=-1.33364…+pin

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2tan^2(x)+9tan(x)+3=0

2 tan^{2} (x) + 9 tan (x) + 3 = 0

x = - 0.34780 \dots + πn, x = - 1.33364 \dots + πn

Degrees

x = - 19.92767 \dots^{\circ} + 18 0^{\circ} n, x = - 76.41251 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 tan^{2} (x) + 9 tan (x) + 3 = 0

Solve by substitution

2 tan^{2} (x) + 9 tan (x) + 3 = 0

Let:

tan (x) = u

2 u^{2} + 9 u + 3 = 0

2 u^{2} + 9 u + 3 = 0 : u = \frac{- 9 + 57}{4}, u = \frac{- 9 - 57}{4}

2 u^{2} + 9 u + 3 = 0

Solve with the quadratic formula

2 u^{2} + 9 u + 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 9, c = 3

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

9^{2} - 4 \cdot 2 \cdot 3 = 57

9^{2} - 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 9^{2} - 24

9^{2} = 81

= 81 - 24

Subtract the numbers:

81 - 24 = 57

= 57

u_{1, 2} = \frac{- 9 \pm 57}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 9 + 57}{2 \cdot 2}, u_{2} = \frac{- 9 - 57}{2 \cdot 2}

u = \frac{- 9 + 57}{2 \cdot 2} : \frac{- 9 + 57}{4}

\frac{- 9 + 57}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 9 + 57}{4}

u = \frac{- 9 - 57}{2 \cdot 2} : \frac{- 9 - 57}{4}

\frac{- 9 - 57}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 9 - 57}{4}

The solutions to the quadratic equation are:

u = \frac{- 9 + 57}{4}, u = \frac{- 9 - 57}{4}

Substitute back

u = tan (x)

tan (x) = \frac{- 9 + 57}{4}, tan (x) = \frac{- 9 - 57}{4}

tan (x) = \frac{- 9 + 57}{4}, tan (x) = \frac{- 9 - 57}{4}

tan (x) = \frac{- 9 + 57}{4} : x = arctan (\frac{- 9 + 57}{4}) + πn

tan (x) = \frac{- 9 + 57}{4}

Apply trig inverse properties

tan (x) = \frac{- 9 + 57}{4}

General solutions for

tan (x) = \frac{- 9 + 57}{4}

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

x = arctan (\frac{- 9 + 57}{4}) + πn

x = arctan (\frac{- 9 + 57}{4}) + πn

tan (x) = \frac{- 9 - 57}{4} : x = arctan (\frac{- 9 - 57}{4}) + πn

tan (x) = \frac{- 9 - 57}{4}

Apply trig inverse properties

tan (x) = \frac{- 9 - 57}{4}

General solutions for

tan (x) = \frac{- 9 - 57}{4}

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

x = arctan (\frac{- 9 - 57}{4}) + πn

x = arctan (\frac{- 9 - 57}{4}) + πn

Combine all the solutions

x = arctan (\frac{- 9 + 57}{4}) + πn, x = arctan (\frac{- 9 - 57}{4}) + πn

Show solutions in decimal form

x = - 0.34780 \dots + πn, x = - 1.33364 \dots + πn

What is the general solution for 2tan^2(x)+9tan(x)+3=0 ?
The general solution for 2tan^2(x)+9tan(x)+3=0 is x=-0.34780…+pin,x=-1.33364…+pin