What is the general solution for tan^2(x)-6tan(x)+1=0 ?

The general solution for tan^2(x)-6tan(x)+1=0 is x=1.40087…+pin,x=0.16991…+pin

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tan^2(x)-6tan(x)+1=0

Solution

tan^{2} (x) - 6 tan (x) + 1 = 0

Solution

x = 1.40087 \dots + πn, x = 0.16991 \dots + πn

Degrees

x = 80.26438 \dots^{\circ} + 18 0^{\circ} n, x = 9.73561 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) - 6 tan (x) + 1 = 0

Solve by substitution

tan^{2} (x) - 6 tan (x) + 1 = 0

Let:

tan (x) = u

u^{2} - 6 u + 1 = 0

u^{2} - 6 u + 1 = 0 : u = 3 + 22, u = 3 - 22

u^{2} - 6 u + 1 = 0

Solve with the quadratic formula

u^{2} - 6 u + 1 = 0

Quadratic Equation Formula:

For

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

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

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

(- 6)^{2} - 4 \cdot 1 \cdot 1 = 42

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

Subtract the numbers:

36 - 4 = 32

= 32

Prime factorization of

32 : 2^{5}

32

32

divides by

232 = 16 \cdot 2

= 2 \cdot 16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{4} \cdot 2

Apply radical rule:

= 2 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

Refine

= 42

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

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 4 2}{2 \cdot 1}, u_{2} = \frac{- ( - 6 ) - 4 2}{2 \cdot 1}

u = \frac{- ( - 6 ) + 4 2}{2 \cdot 1} : 3 + 22

\frac{- ( - 6 ) + 4 2}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 + 4 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

6 + 42 : 2 (3 + 22)

6 + 42

Rewrite as

= 2 \cdot 3 + 2 \cdot 22

Factor out common term

2

= 2 (3 + 22)

= \frac{2 ( 3 + 2 2 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 22

u = \frac{- ( - 6 ) - 4 2}{2 \cdot 1} : 3 - 22

\frac{- ( - 6 ) - 4 2}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{6 - 4 2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

6 - 42 : 2 (3 - 22)

6 - 42

Rewrite as

= 2 \cdot 3 - 2 \cdot 22

Factor out common term

2

= 2 (3 - 22)

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

Divide the numbers:

\frac{2}{2} = 1

= 3 - 22

The solutions to the quadratic equation are:

u = 3 + 22, u = 3 - 22

Substitute back

u = tan (x)

tan (x) = 3 + 22, tan (x) = 3 - 22

tan (x) = 3 + 22, tan (x) = 3 - 22

tan (x) = 3 + 22 : x = arctan (3 + 22) + πn

tan (x) = 3 + 22

Apply trig inverse properties

tan (x) = 3 + 22

General solutions for

tan (x) = 3 + 22

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

x = arctan (3 + 22) + πn

x = arctan (3 + 22) + πn

tan (x) = 3 - 22 : x = arctan (3 - 22) + πn

tan (x) = 3 - 22

Apply trig inverse properties

tan (x) = 3 - 22

General solutions for

tan (x) = 3 - 22

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

x = arctan (3 - 22) + πn

x = arctan (3 - 22) + πn

Combine all the solutions

x = arctan (3 + 22) + πn, x = arctan (3 - 22) + πn

Show solutions in decimal form

x = 1.40087 \dots + πn, x = 0.16991 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)-6tan(x)+1=0 ?
The general solution for tan^2(x)-6tan(x)+1=0 is x=1.40087…+pin,x=0.16991…+pin