What is the general solution for 4tan^2(x)+5tan(x)=6 ?

The general solution for 4tan^2(x)+5tan(x)=6 is x=0.64350…+pin,x=-1.10714…+pin

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Popular >

4tan^2(x)+5tan(x)=6

Solution

4 tan^{2} (x) + 5 tan (x) = 6

Solution

x = 0.64350 \dots + πn, x = - 1.10714 \dots + πn

Degrees

x = 36.86989 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) + 5 tan (x) = 6

Solve by substitution

4 tan^{2} (x) + 5 tan (x) = 6

Let:

tan (x) = u

4 u^{2} + 5 u = 6

4 u^{2} + 5 u = 6 : u = \frac{3}{4}, u = - 2

4 u^{2} + 5 u = 6

Move

6

to the left side

4 u^{2} + 5 u = 6

Subtract

6

from both sides

4 u^{2} + 5 u - 6 = 6 - 6

Simplify

4 u^{2} + 5 u - 6 = 0

4 u^{2} + 5 u - 6 = 0

Solve with the quadratic formula

4 u^{2} + 5 u - 6 = 0

Quadratic Equation Formula:

For

a = 4, b = 5, c = - 6

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

5^{2} - 4 \cdot 4 (- 6) = 11

5^{2} - 4 \cdot 4 (- 6)

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 4 \cdot 6

Multiply the numbers:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Add the numbers:

25 + 96 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 5 \pm 11}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 5 + 11}{2 \cdot 4}, u_{2} = \frac{- 5 - 11}{2 \cdot 4}

u = \frac{- 5 + 11}{2 \cdot 4} : \frac{3}{4}

\frac{- 5 + 11}{2 \cdot 4}

Add/Subtract the numbers:

- 5 + 11 = 6

= \frac{6}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{6}{8}

Cancel the common factor:

2

= \frac{3}{4}

u = \frac{- 5 - 11}{2 \cdot 4} : - 2

\frac{- 5 - 11}{2 \cdot 4}

Subtract the numbers:

- 5 - 11 = - 16

= \frac{- 16}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 16}{8}

Apply the fraction rule:

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

= - \frac{16}{8}

Divide the numbers:

\frac{16}{8} = 2

= - 2

The solutions to the quadratic equation are:

u = \frac{3}{4}, u = - 2

Substitute back

u = tan (x)

tan (x) = \frac{3}{4}, tan (x) = - 2

tan (x) = \frac{3}{4}, tan (x) = - 2

tan (x) = \frac{3}{4} : x = arctan (\frac{3}{4}) + πn

tan (x) = \frac{3}{4}

Apply trig inverse properties

tan (x) = \frac{3}{4}

General solutions for

tan (x) = \frac{3}{4}

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

x = arctan (\frac{3}{4}) + πn

x = arctan (\frac{3}{4}) + πn

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

Combine all the solutions

x = arctan (\frac{3}{4}) + πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = 0.64350 \dots + πn, x = - 1.10714 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)+5tan(x)=6 ?
The general solution for 4tan^2(x)+5tan(x)=6 is x=0.64350…+pin,x=-1.10714…+pin