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

The general solution for tan^2(x)+3tan(x)-5=0 is x=0.87300…+pin,x=-1.33665…+pin

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

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

x = 0.87300 \dots + πn, x = - 1.33665 \dots + πn

Degrees

x = 50.01961 \dots^{\circ} + 18 0^{\circ} n, x = - 76.58466 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

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

u^{2} + 3 u - 5 = 0 : u = \frac{- 3 + 29}{2}, u = \frac{- 3 - 29}{2}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = 3, c = - 5

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

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

3^{2} - 4 \cdot 1 \cdot (- 5) = 29

3^{2} - 4 \cdot 1 \cdot (- 5)

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 1 \cdot 5 = 20

= 3^{2} + 20

3^{2} = 9

= 9 + 20

Add the numbers:

9 + 20 = 29

= 29

u_{1, 2} = \frac{- 3 \pm 29}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 29}{2 \cdot 1}, u_{2} = \frac{- 3 - 29}{2 \cdot 1}

u = \frac{- 3 + 29}{2 \cdot 1} : \frac{- 3 + 29}{2}

\frac{- 3 + 29}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 29}{2}

u = \frac{- 3 - 29}{2 \cdot 1} : \frac{- 3 - 29}{2}

\frac{- 3 - 29}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 29}{2}

The solutions to the quadratic equation are:

u = \frac{- 3 + 29}{2}, u = \frac{- 3 - 29}{2}

Substitute back

u = tan (x)

tan (x) = \frac{- 3 + 29}{2}, tan (x) = \frac{- 3 - 29}{2}

tan (x) = \frac{- 3 + 29}{2}, tan (x) = \frac{- 3 - 29}{2}

tan (x) = \frac{- 3 + 29}{2} : x = arctan (\frac{- 3 + 29}{2}) + πn

tan (x) = \frac{- 3 + 29}{2}

Apply trig inverse properties

tan (x) = \frac{- 3 + 29}{2}

General solutions for

tan (x) = \frac{- 3 + 29}{2}

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

x = arctan (\frac{- 3 + 29}{2}) + πn

x = arctan (\frac{- 3 + 29}{2}) + πn

tan (x) = \frac{- 3 - 29}{2} : x = arctan (\frac{- 3 - 29}{2}) + πn

tan (x) = \frac{- 3 - 29}{2}

Apply trig inverse properties

tan (x) = \frac{- 3 - 29}{2}

General solutions for

tan (x) = \frac{- 3 - 29}{2}

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

x = arctan (\frac{- 3 - 29}{2}) + πn

x = arctan (\frac{- 3 - 29}{2}) + πn

Combine all the solutions

x = arctan (\frac{- 3 + 29}{2}) + πn, x = arctan (\frac{- 3 - 29}{2}) + πn

Show solutions in decimal form

x = 0.87300 \dots + πn, x = - 1.33665 \dots + πn

tan (\frac{1}{2} x) - 3 = 0

2 cos (2 θ) = 6 cos (θ) - 4

5 sin (θ) - 4 = 2 sin (θ) - 6

cos (2 θ) - cos (4 θ) = 0

tan (x) = 1.4

What is the general solution for tan^2(x)+3tan(x)-5=0 ?
The general solution for tan^2(x)+3tan(x)-5=0 is x=0.87300…+pin,x=-1.33665…+pin