What is the general solution for tan^2(x)+3tan(x)-10=0 ?

The general solution for tan^2(x)+3tan(x)-10=0 is x=1.10714…+pin,x=-1.37340…+pin

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tan^2(x)+3tan(x)-10=0

Solution

tan^{2} (x) + 3 tan (x) - 10 = 0

Solution

x = 1.10714 \dots + πn, x = - 1.37340 \dots + πn

Degrees

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = - 78.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) + 3 tan (x) - 10 = 0

Solve by substitution

tan^{2} (x) + 3 tan (x) - 10 = 0

Let:

tan (x) = u

u^{2} + 3 u - 10 = 0

u^{2} + 3 u - 10 = 0 : u = 2, u = - 5

u^{2} + 3 u - 10 = 0

Solve with the quadratic formula

u^{2} + 3 u - 10 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = - 10

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 10 )}{2 \cdot 1}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 1 \cdot ( - 10 )}{2 \cdot 1}

3^{2} - 4 \cdot 1 \cdot (- 10) = 7

3^{2} - 4 \cdot 1 \cdot (- 10)

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 1 \cdot 10

Multiply the numbers:

4 \cdot 1 \cdot 10 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

Add the numbers:

9 + 40 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 3 \pm 7}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 7}{2 \cdot 1}, u_{2} = \frac{- 3 - 7}{2 \cdot 1}

u = \frac{- 3 + 7}{2 \cdot 1} : 2

\frac{- 3 + 7}{2 \cdot 1}

Add/Subtract the numbers:

- 3 + 7 = 4

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u = \frac{- 3 - 7}{2 \cdot 1} : - 5

\frac{- 3 - 7}{2 \cdot 1}

Subtract the numbers:

- 3 - 7 = - 10

= \frac{- 10}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10}{2}

Apply the fraction rule:

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

= - \frac{10}{2}

Divide the numbers:

\frac{10}{2} = 5

= - 5

The solutions to the quadratic equation are:

u = 2, u = - 5

Substitute back

u = tan (x)

tan (x) = 2, tan (x) = - 5

tan (x) = 2, tan (x) = - 5

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

tan (x) = - 5 : x = arctan (- 5) + πn

tan (x) = - 5

Apply trig inverse properties

tan (x) = - 5

General solutions for

tan (x) = - 5

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

x = arctan (- 5) + πn

x = arctan (- 5) + πn

Combine all the solutions

x = arctan (2) + πn, x = arctan (- 5) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = - 1.37340 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+3tan(x)-10=0 ?
The general solution for tan^2(x)+3tan(x)-10=0 is x=1.10714…+pin,x=-1.37340…+pin