What is the general solution for tan^2(x)-3tan(x)-28=0 ?

The general solution for tan^2(x)-3tan(x)-28=0 is x=1.42889…+pin,x=-1.32581…+pin

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tan^2(x)-3tan(x)-28=0

Solution

tan^{2} (x) - 3 tan (x) - 28 = 0

Solution

x = 1.42889 \dots + πn, x = - 1.32581 \dots + πn

Degrees

x = 81.86989 \dots^{\circ} + 18 0^{\circ} n, x = - 75.96375 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (x) - 3 tan (x) - 28 = 0

Solve by substitution

tan^{2} (x) - 3 tan (x) - 28 = 0

Let:

tan (x) = u

u^{2} - 3 u - 28 = 0

u^{2} - 3 u - 28 = 0 : u = 7, u = - 4

u^{2} - 3 u - 28 = 0

Solve with the quadratic formula

u^{2} - 3 u - 28 = 0

Quadratic Equation Formula:

For

a = 1, b = - 3, c = - 28

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot ( - 28 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot ( - 28 )}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot (- 28) = 11

(- 3)^{2} - 4 \cdot 1 \cdot (- 28)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 28

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 1 \cdot 28

Multiply the numbers:

4 \cdot 1 \cdot 28 = 112

= 3^{2} + 112

3^{2} = 9

= 9 + 112

Add the numbers:

9 + 112 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 3 ) \pm 11}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 11}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 11}{2 \cdot 1}

u = \frac{- ( - 3 ) + 11}{2 \cdot 1} : 7

\frac{- ( - 3 ) + 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 11}{2 \cdot 1}

Add the numbers:

3 + 11 = 14

= \frac{14}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{14}{2}

Divide the numbers:

\frac{14}{2} = 7

= 7

u = \frac{- ( - 3 ) - 11}{2 \cdot 1} : - 4

\frac{- ( - 3 ) - 11}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 11}{2 \cdot 1}

Subtract the numbers:

3 - 11 = - 8

= \frac{- 8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8}{2}

Apply the fraction rule:

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

= - \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= - 4

The solutions to the quadratic equation are:

u = 7, u = - 4

Substitute back

u = tan (x)

tan (x) = 7, tan (x) = - 4

tan (x) = 7, tan (x) = - 4

tan (x) = 7 : x = arctan (7) + πn

tan (x) = 7

Apply trig inverse properties

tan (x) = 7

General solutions for

tan (x) = 7

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

x = arctan (7) + πn

x = arctan (7) + πn

tan (x) = - 4 : x = arctan (- 4) + πn

tan (x) = - 4

Apply trig inverse properties

tan (x) = - 4

General solutions for

tan (x) = - 4

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

x = arctan (- 4) + πn

x = arctan (- 4) + πn

Combine all the solutions

x = arctan (7) + πn, x = arctan (- 4) + πn

Show solutions in decimal form

x = 1.42889 \dots + πn, x = - 1.32581 \dots + πn

Popular Examples

cot(x)=-0.69

cot (x) = - 0.69

sin(θ)=-1/8

sin (θ) = - \frac{1}{8}

0=2cos^2(θ)-cos(θ)-1

0 = 2 cos^{2} (θ) - cos (θ) - 1

sin(x)=(sqrt(3))/(1.65)

sin (x) = \frac{3}{1.65}

tan(x)=-2,tan(-x)

tan (x) = - 2, tan (- x)

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)-3tan(x)-28=0 ?
The general solution for tan^2(x)-3tan(x)-28=0 is x=1.42889…+pin,x=-1.32581…+pin