What is the general solution for 4.718tan^2(θ)-13.75tan(θ)+8.018=0 ?

The general solution for 4.718tan^2(θ)-13.75tan(θ)+8.018=0 is θ=1.12790…+pin,θ=0.67843…+pin

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4.718tan^2(θ)-13.75tan(θ)+8.018=0

Solution

4.718 tan^{2} (θ) - 13.75 tan (θ) + 8.018 = 0

Solution

θ = 1.12790 \dots + πn, θ = 0.67843 \dots + πn

Degrees

θ = 64.62419 \dots^{\circ} + 18 0^{\circ} n, θ = 38.87154 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4.718 tan^{2} (θ) - 13.75 tan (θ) + 8.018 = 0

Solve by substitution

4.718 tan^{2} (θ) - 13.75 tan (θ) + 8.018 = 0

Let:

tan (θ) = u

4.718 u^{2} - 13.75 u + 8.018 = 0

4.718 u^{2} - 13.75 u + 8.018 = 0 : u = \frac{6875 + 9436701}{4718}, u = \frac{6875 - 9436701}{4718}

4.718 u^{2} - 13.75 u + 8.018 = 0

Multiply both sides by

1000

4.718 u^{2} - 13.75 u + 8.018 = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

3

digits to the right of the decimal point, therefore multiply by

1000

4.718 u^{2} \cdot 1000 - 13.75 u \cdot 1000 + 8.018 \cdot 1000 = 0 \cdot 1000

Refine

4718 u^{2} - 13750 u + 8018 = 0

4718 u^{2} - 13750 u + 8018 = 0

Solve with the quadratic formula

4718 u^{2} - 13750 u + 8018 = 0

Quadratic Equation Formula:

For

a = 4718, b = - 13750, c = 8018

u_{1, 2} = \frac{- ( - 13750 ) \pm ( - 13750 ) ^{2} - 4 \cdot 4718 \cdot 8018}{2 \cdot 4718}

u_{1, 2} = \frac{- ( - 13750 ) \pm ( - 13750 ) ^{2} - 4 \cdot 4718 \cdot 8018}{2 \cdot 4718}

(- 13750)^{2} - 4 \cdot 4718 \cdot 8018 = 29436701

(- 13750)^{2} - 4 \cdot 4718 \cdot 8018

Apply exponent rule:

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

n

is even

(- 13750)^{2} = 1375 0^{2}

= 1375 0^{2} - 4 \cdot 4718 \cdot 8018

Multiply the numbers:

4 \cdot 4718 \cdot 8018 = 151315696

= 1375 0^{2} - 151315696

1375 0^{2} = 189062500

= 189062500 - 151315696

Subtract the numbers:

189062500 - 151315696 = 37746804

= 37746804

Prime factorization of

37746804 : 2^{2} \cdot 3 \cdot 179 \cdot 17573

37746804

= 2^{2} \cdot 3 \cdot 179 \cdot 17573

Apply radical rule:

= 2^{2} 3 \cdot 179 \cdot 17573

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 179 \cdot 17573

Refine

= 29436701

u_{1, 2} = \frac{- ( - 13750 ) \pm 2 9436701}{2 \cdot 4718}

Separate the solutions

u_{1} = \frac{- ( - 13750 ) + 2 9436701}{2 \cdot 4718}, u_{2} = \frac{- ( - 13750 ) - 2 9436701}{2 \cdot 4718}

u = \frac{- ( - 13750 ) + 2 9436701}{2 \cdot 4718} : \frac{6875 + 9436701}{4718}

\frac{- ( - 13750 ) + 2 9436701}{2 \cdot 4718}

Apply rule

- (- a) = a

= \frac{13750 + 2 9436701}{2 \cdot 4718}

Multiply the numbers:

2 \cdot 4718 = 9436

= \frac{13750 + 2 9436701}{9436}

Factor

13750 + 29436701 : 2 (6875 + 9436701)

13750 + 29436701

Rewrite as

= 2 \cdot 6875 + 29436701

Factor out common term

2

= 2 (6875 + 9436701)

= \frac{2 ( 6875 + 9436701 )}{9436}

Cancel the common factor:

2

= \frac{6875 + 9436701}{4718}

u = \frac{- ( - 13750 ) - 2 9436701}{2 \cdot 4718} : \frac{6875 - 9436701}{4718}

\frac{- ( - 13750 ) - 2 9436701}{2 \cdot 4718}

Apply rule

- (- a) = a

= \frac{13750 - 2 9436701}{2 \cdot 4718}

Multiply the numbers:

2 \cdot 4718 = 9436

= \frac{13750 - 2 9436701}{9436}

Factor

13750 - 29436701 : 2 (6875 - 9436701)

13750 - 29436701

Rewrite as

= 2 \cdot 6875 - 29436701

Factor out common term

2

= 2 (6875 - 9436701)

= \frac{2 ( 6875 - 9436701 )}{9436}

Cancel the common factor:

2

= \frac{6875 - 9436701}{4718}

The solutions to the quadratic equation are:

u = \frac{6875 + 9436701}{4718}, u = \frac{6875 - 9436701}{4718}

Substitute back

u = tan (θ)

tan (θ) = \frac{6875 + 9436701}{4718}, tan (θ) = \frac{6875 - 9436701}{4718}

tan (θ) = \frac{6875 + 9436701}{4718}, tan (θ) = \frac{6875 - 9436701}{4718}

tan (θ) = \frac{6875 + 9436701}{4718} : θ = arctan (\frac{6875 + 9436701}{4718}) + πn

tan (θ) = \frac{6875 + 9436701}{4718}

Apply trig inverse properties

tan (θ) = \frac{6875 + 9436701}{4718}

General solutions for

tan (θ) = \frac{6875 + 9436701}{4718}

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

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

θ = arctan (\frac{6875 + 9436701}{4718}) + πn

tan (θ) = \frac{6875 - 9436701}{4718} : θ = arctan (\frac{6875 - 9436701}{4718}) + πn

tan (θ) = \frac{6875 - 9436701}{4718}

Apply trig inverse properties

tan (θ) = \frac{6875 - 9436701}{4718}

General solutions for

tan (θ) = \frac{6875 - 9436701}{4718}

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

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

θ = arctan (\frac{6875 - 9436701}{4718}) + πn

Combine all the solutions

θ = arctan (\frac{6875 + 9436701}{4718}) + πn, θ = arctan (\frac{6875 - 9436701}{4718}) + πn

Show solutions in decimal form

θ = 1.12790 \dots + πn, θ = 0.67843 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4.718tan^2(θ)-13.75tan(θ)+8.018=0 ?
The general solution for 4.718tan^2(θ)-13.75tan(θ)+8.018=0 is θ=1.12790…+pin,θ=0.67843…+pin