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

The general solution for tan^2(x)+4tan(x)+2=0 is x=-0.52990…+pin,x=-1.28587…+pin

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Popular >

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

Solution

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

Solution

x = - 0.52990 \dots + πn, x = - 1.28587 \dots + πn

Degrees

x = - 30.36119 \dots^{\circ} + 18 0^{\circ} n, x = - 73.67505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

u^{2} + 4 u + 2 = 0 : u = - 2 + 2, u = - 2 - 2

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = 4, c = 2

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

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

4^{2} - 4 \cdot 1 \cdot 2 = 22

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 4^{2} - 8

4^{2} = 16

= 16 - 8

Subtract the numbers:

16 - 8 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

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

Separate the solutions

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

u = \frac{- 4 + 2 2}{2 \cdot 1} : - 2 + 2

\frac{- 4 + 2 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

- 4 + 22 : 2 (- 2 + 2)

- 4 + 22

Rewrite as

= - 2 \cdot 2 + 22

Factor out common term

2

= 2 (- 2 + 2)

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

Divide the numbers:

\frac{2}{2} = 1

= - 2 + 2

u = \frac{- 4 - 2 2}{2 \cdot 1} : - 2 - 2

\frac{- 4 - 2 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

- 4 - 22 : - 2 (2 + 2)

- 4 - 22

Rewrite as

= - 2 \cdot 2 - 22

Factor out common term

2

= - 2 (2 + 2)

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

Divide the numbers:

\frac{2}{2} = 1

= - (2 + 2)

Negate

- (2 + 2) = - 2 - 2

= - 2 - 2

The solutions to the quadratic equation are:

u = - 2 + 2, u = - 2 - 2

Substitute back

u = tan (x)

tan (x) = - 2 + 2, tan (x) = - 2 - 2

tan (x) = - 2 + 2, tan (x) = - 2 - 2

tan (x) = - 2 + 2 : x = arctan (- 2 + 2) + πn

tan (x) = - 2 + 2

Apply trig inverse properties

tan (x) = - 2 + 2

General solutions for

tan (x) = - 2 + 2

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

x = arctan (- 2 + 2) + πn

x = arctan (- 2 + 2) + πn

tan (x) = - 2 - 2 : x = arctan (- 2 - 2) + πn

tan (x) = - 2 - 2

Apply trig inverse properties

tan (x) = - 2 - 2

General solutions for

tan (x) = - 2 - 2

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

x = arctan (- 2 - 2) + πn

x = arctan (- 2 - 2) + πn

Combine all the solutions

x = arctan (- 2 + 2) + πn, x = arctan (- 2 - 2) + πn

Show solutions in decimal form

x = - 0.52990 \dots + πn, x = - 1.28587 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+4tan(x)+2=0 ?
The general solution for tan^2(x)+4tan(x)+2=0 is x=-0.52990…+pin,x=-1.28587…+pin