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

The general solution for 5tan(x)+5tan^2(x)=0 is x=pin,x=(3pi)/4+pin

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

Solution

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

Solution

x = πn, x = \frac{3 π}{4} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

5 u + 5 u^{2} = 0

5 u + 5 u^{2} = 0 : u = 0, u = - 1

5 u + 5 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

5 u^{2} + 5 u = 0

Solve with the quadratic formula

5 u^{2} + 5 u = 0

Quadratic Equation Formula:

For

a = 5, b = 5, c = 0

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 5 \cdot 0}{2 \cdot 5}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 5 \cdot 0}{2 \cdot 5}

5^{2} - 4 \cdot 5 \cdot 0 = 5

5^{2} - 4 \cdot 5 \cdot 0

Apply rule

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 5

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

Separate the solutions

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

u = \frac{- 5 + 5}{2 \cdot 5} : 0

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

Add/Subtract the numbers:

- 5 + 5 = 0

= \frac{0}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{0}{10}

Apply rule

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

= 0

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

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

Subtract the numbers:

- 5 - 5 = - 10

= \frac{- 10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{10}

Apply the fraction rule:

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

= - \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = 0, u = - 1

Substitute back

u = tan (x)

tan (x) = 0, tan (x) = - 1

tan (x) = 0, tan (x) = - 1

tan (x) = 0 : x = πn

tan (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

tan (x) = - 1 : x = \frac{3 π}{4} + πn

tan (x) = - 1

General solutions for

tan (x) = - 1

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 = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = πn, x = \frac{3 π}{4} + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5tan(x)+5tan^2(x)=0 ?
The general solution for 5tan(x)+5tan^2(x)=0 is x=pin,x=(3pi)/4+pin