What is the general solution for tan^2(x)+3tan(x)=-1 ?

The general solution for tan^2(x)+3tan(x)=-1 is x=-0.36486…+pin,x=-1.20593…+pin

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Popular >

tan^2(x)+3tan(x)=-1

Solution

tan^{2} (x) + 3 tan (x) = - 1

Solution

x = - 0.36486 \dots + πn, x = - 1.20593 \dots + πn

Degrees

x = - 20.90515 \dots^{\circ} + 18 0^{\circ} n, x = - 69.09484 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + 3 tan (x) = - 1

Solve by substitution

tan^{2} (x) + 3 tan (x) = - 1

Let:

tan (x) = u

u^{2} + 3 u = - 1

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

u^{2} + 3 u = - 1

Move

1

to the left side

u^{2} + 3 u = - 1

Add

1

to both sides

u^{2} + 3 u + 1 = - 1 + 1

Simplify

u^{2} + 3 u + 1 = 0

u^{2} + 3 u + 1 = 0

Solve with the quadratic formula

u^{2} + 3 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = 1

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

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

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

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

Subtract the numbers:

9 - 4 = 5

= 5

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

Separate the solutions

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

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

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 5}{2}

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

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 5}{2}

The solutions to the quadratic equation are:

u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

Substitute back

u = tan (x)

tan (x) = \frac{- 3 + 5}{2}, tan (x) = \frac{- 3 - 5}{2}

tan (x) = \frac{- 3 + 5}{2}, tan (x) = \frac{- 3 - 5}{2}

tan (x) = \frac{- 3 + 5}{2} : x = arctan (\frac{- 3 + 5}{2}) + πn

tan (x) = \frac{- 3 + 5}{2}

Apply trig inverse properties

tan (x) = \frac{- 3 + 5}{2}

General solutions for

tan (x) = \frac{- 3 + 5}{2}

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

x = arctan (\frac{- 3 + 5}{2}) + πn

x = arctan (\frac{- 3 + 5}{2}) + πn

tan (x) = \frac{- 3 - 5}{2} : x = arctan (\frac{- 3 - 5}{2}) + πn

tan (x) = \frac{- 3 - 5}{2}

Apply trig inverse properties

tan (x) = \frac{- 3 - 5}{2}

General solutions for

tan (x) = \frac{- 3 - 5}{2}

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

x = arctan (\frac{- 3 - 5}{2}) + πn

x = arctan (\frac{- 3 - 5}{2}) + πn

Combine all the solutions

x = arctan (\frac{- 3 + 5}{2}) + πn, x = arctan (\frac{- 3 - 5}{2}) + πn

Show solutions in decimal form

x = - 0.36486 \dots + πn, x = - 1.20593 \dots + πn

Popular Examples

7sin(x)-4=0

7 sin (x) - 4 = 0

(sin(x)-2)(cos(x)+7)=0

(sin (x) - 2) (cos (x) + 7) = 0

-4tan(θ)-10=4tan(θ)-7

- 4 tan (θ) - 10 = 4 tan (θ) - 7

sin(x)=sin(x^2)

sin (x) = sin (x^{2})

sin(x)=-0.23

sin (x) = - 0.23

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+3tan(x)=-1 ?
The general solution for tan^2(x)+3tan(x)=-1 is x=-0.36486…+pin,x=-1.20593…+pin