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

The general solution for 6tan(x)=tan^2(x),x=0 is x=0

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6tan(x)=tan^2(x),x=0

Solution

6 tan (x) = tan^{2} (x), x = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

6 tan (x) = tan^{2} (x), x = 0

Solve by substitution

6 tan (x) = tan^{2} (x)

Let:

tan (x) = u

6 u = u^{2}

6 u = u^{2} : u = 6, u = 0

6 u = u^{2}

Switch sides

u^{2} = 6 u

Move

6 u

to the left side

u^{2} = 6 u

Subtract

6 u

from both sides

u^{2} - 6 u = 6 u - 6 u

Simplify

u^{2} - 6 u = 0

u^{2} - 6 u = 0

Solve with the quadratic formula

u^{2} - 6 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 6, c = 0

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 6)^{2} - 4 \cdot 1 \cdot 0 = 6

(- 6)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 6^{2} - 0

6^{2} - 0 = 6^{2}

= 6^{2}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 6

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

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 6}{2 \cdot 1}, u_{2} = \frac{- ( - 6 ) - 6}{2 \cdot 1}

u = \frac{- ( - 6 ) + 6}{2 \cdot 1} : 6

\frac{- ( - 6 ) + 6}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 + 6}{2 \cdot 1}

Add the numbers:

6 + 6 = 12

= \frac{12}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6

u = \frac{- ( - 6 ) - 6}{2 \cdot 1} : 0

\frac{- ( - 6 ) - 6}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 - 6}{2 \cdot 1}

Subtract the numbers:

6 - 6 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 6, u = 0

Substitute back

u = tan (x)

tan (x) = 6, tan (x) = 0

tan (x) = 6, tan (x) = 0

tan (x) = 6, x = 0 :

No Solution

tan (x) = 6, x = 0

Apply trig inverse properties

tan (x) = 6

General solutions for

tan (x) = 6

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

x = arctan (6) + πn

x = arctan (6) + πn

Solutions for the range

x = 0

No Solution

tan (x) = 0, x = 0 : x = 0

tan (x) = 0, x = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

Solutions for the range

x = 0

x = 0

Combine all the solutions

x = 0

Graph

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Frequently Asked Questions (FAQ)

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