What is the general solution for 6tan^2(x)+13tan(x)+6=0 ?

The general solution for 6tan^2(x)+13tan(x)+6=0 is x=-0.58800…+pin,x=-0.98279…+pin

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Popular Trigonometry >

6tan^2(x)+13tan(x)+6=0

Solution

6 tan^{2} (x) + 13 tan (x) + 6 = 0

Solution

x = - 0.58800 \dots + πn, x = - 0.98279 \dots + πn

Degrees

x = - 33.69006 \dots^{\circ} + 18 0^{\circ} n, x = - 56.30993 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 tan^{2} (x) + 13 tan (x) + 6 = 0

Solve by substitution

6 tan^{2} (x) + 13 tan (x) + 6 = 0

Let:

tan (x) = u

6 u^{2} + 13 u + 6 = 0

6 u^{2} + 13 u + 6 = 0 : u = - \frac{2}{3}, u = - \frac{3}{2}

6 u^{2} + 13 u + 6 = 0

Solve with the quadratic formula

6 u^{2} + 13 u + 6 = 0

Quadratic Equation Formula:

For

a = 6, b = 13, c = 6

u_{1, 2} = \frac{- 13 \pm 1 3 ^{2} - 4 \cdot 6 \cdot 6}{2 \cdot 6}

u_{1, 2} = \frac{- 13 \pm 1 3 ^{2} - 4 \cdot 6 \cdot 6}{2 \cdot 6}

1 3^{2} - 4 \cdot 6 \cdot 6 = 5

1 3^{2} - 4 \cdot 6 \cdot 6

Multiply the numbers:

4 \cdot 6 \cdot 6 = 144

= 1 3^{2} - 144

1 3^{2} = 169

= 169 - 144

Subtract the numbers:

169 - 144 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 13 \pm 5}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- 13 + 5}{2 \cdot 6}, u_{2} = \frac{- 13 - 5}{2 \cdot 6}

u = \frac{- 13 + 5}{2 \cdot 6} : - \frac{2}{3}

\frac{- 13 + 5}{2 \cdot 6}

Add/Subtract the numbers:

- 13 + 5 = - 8

= \frac{- 8}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 8}{12}

Apply the fraction rule:

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

= - \frac{8}{12}

Cancel the common factor:

4

= - \frac{2}{3}

u = \frac{- 13 - 5}{2 \cdot 6} : - \frac{3}{2}

\frac{- 13 - 5}{2 \cdot 6}

Subtract the numbers:

- 13 - 5 = - 18

= \frac{- 18}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 18}{12}

Apply the fraction rule:

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

= - \frac{18}{12}

Cancel the common factor:

6

= - \frac{3}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = tan (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{2}{3}) + πn

x = arctan (- \frac{2}{3}) + πn

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

Combine all the solutions

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

Show solutions in decimal form

x = - 0.58800 \dots + πn, x = - 0.98279 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6tan^2(x)+13tan(x)+6=0 ?
The general solution for 6tan^2(x)+13tan(x)+6=0 is x=-0.58800…+pin,x=-0.98279…+pin