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Popular >

3cos^2(x)+4cos^4(x)-5=0

Solution

3 cos^{2} (x) + 4 cos^{4} (x) - 5 = 0

Solution

x = 0.45831 \dots + 2 πn, x = 2 π - 0.45831 \dots + 2 πn, x = 2.68327 \dots + 2 πn, x = - 2.68327 \dots + 2 πn

Degrees

x = 26.25960 \dots^{\circ} + 36 0^{\circ} n, x = 333.74039 \dots^{\circ} + 36 0^{\circ} n, x = 153.74039 \dots^{\circ} + 36 0^{\circ} n, x = - 153.74039 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos^{2} (x) + 4 cos^{4} (x) - 5 = 0

Solve by substitution

3 cos^{2} (x) + 4 cos^{4} (x) - 5 = 0

Let:

cos (x) = u

3 u^{2} + 4 u^{4} - 5 = 0

3 u^{2} + 4 u^{4} - 5 = 0 : u = \frac{2 - 3 + 89}{4}, u = - \frac{2 - 3 + 89}{4}, u = i \frac{2 3 + 89}{4}, u = - i \frac{2 3 + 89}{4}

3 u^{2} + 4 u^{4} - 5 = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

4 u^{4} + 3 u^{2} - 5 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

4 v^{2} + 3 v - 5 = 0

Solve

4 v^{2} + 3 v - 5 = 0 : v = \frac{- 3 + 89}{8}, v = \frac{- 3 - 89}{8}

4 v^{2} + 3 v - 5 = 0

Solve with the quadratic formula

4 v^{2} + 3 v - 5 = 0

Quadratic Equation Formula:

For

a = 4, b = 3, c = - 5

v_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 4 ( - 5 )}{2 \cdot 4}

v_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 4 ( - 5 )}{2 \cdot 4}

3^{2} - 4 \cdot 4 (- 5) = 89

3^{2} - 4 \cdot 4 (- 5)

Apply rule

- (- a) = a

= 3^{2} + 4 \cdot 4 \cdot 5

Multiply the numbers:

4 \cdot 4 \cdot 5 = 80

= 3^{2} + 80

3^{2} = 9

= 9 + 80

Add the numbers:

9 + 80 = 89

= 89

v_{1, 2} = \frac{- 3 \pm 89}{2 \cdot 4}

Separate the solutions

v_{1} = \frac{- 3 + 89}{2 \cdot 4}, v_{2} = \frac{- 3 - 89}{2 \cdot 4}

v = \frac{- 3 + 89}{2 \cdot 4} : \frac{- 3 + 89}{8}

\frac{- 3 + 89}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 3 + 89}{8}

v = \frac{- 3 - 89}{2 \cdot 4} : \frac{- 3 - 89}{8}

\frac{- 3 - 89}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 3 - 89}{8}

The solutions to the quadratic equation are:

v = \frac{- 3 + 89}{8}, v = \frac{- 3 - 89}{8}

v = \frac{- 3 + 89}{8}, v = \frac{- 3 - 89}{8}

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{- 3 + 89}{8} : u = \frac{2 - 3 + 89}{4}, u = - \frac{2 - 3 + 89}{4}

u^{2} = \frac{- 3 + 89}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{- 3 + 89}{8}, u = - \frac{- 3 + 89}{8}

\frac{- 3 + 89}{8} = \frac{2 - 3 + 89}{4}

\frac{- 3 + 89}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{- 3 + 89}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{89 - 3}{2 2}

Rationalize

\frac{- 3 + 89}{2 2} : \frac{2 89 - 3}{4}

\frac{- 3 + 89}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{- 3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 - 3 + 89}{4}

= \frac{2 89 - 3}{4}

= \frac{2 - 3 + 89}{4}

- \frac{- 3 + 89}{8} = - \frac{2 - 3 + 89}{4}

- \frac{- 3 + 89}{8}

Simplify

\frac{- 3 + 89}{8} : \frac{- 3 + 89}{2 2}

\frac{- 3 + 89}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{- 3 + 89}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{89 - 3}{2 2}

= - \frac{89 - 3}{2 2}

Rationalize

- \frac{89 - 3}{2 2} : - \frac{2 89 - 3}{4}

- \frac{89 - 3}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{- 3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 - 3 + 89}{4}

= - \frac{2 89 - 3}{4}

= - \frac{2 - 3 + 89}{4}

u = \frac{2 - 3 + 89}{4}, u = - \frac{2 - 3 + 89}{4}

Solve

u^{2} = \frac{- 3 - 89}{8} : u = i \frac{2 3 + 89}{4}, u = - i \frac{2 3 + 89}{4}

u^{2} = \frac{- 3 - 89}{8}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{- 3 - 89}{8}, u = - \frac{- 3 - 89}{8}

Simplify

\frac{- 3 - 89}{8} : i \frac{2 3 + 89}{4}

\frac{- 3 - 89}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{- 3 - 89}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{- 3 - 89}{2 2}

Apply imaginary number rule:

- a = i a

= \frac{i 3 + 89}{2 2}

Rationalize

\frac{i 3 + 89}{2 2} : \frac{2 i 3 + 89}{4}

\frac{i 3 + 89}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{i 3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 i 3 + 89}{4}

= \frac{2 i 3 + 89}{4}

Rewrite

\frac{2 i 3 + 89}{4}

in standard complex form:

\frac{2 3 + 89}{4} i

\frac{2 i 3 + 89}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 i 3 + 89}{2 ^{2}}

Cancel

\frac{2 i 3 + 89}{2 ^{2}} : \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

\frac{2 i 3 + 89}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} i 3 + 89}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{i 3 + 89}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

= \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= \frac{i 3 + 89}{2 2}

\frac{3 + 89}{2 2} = \frac{2 3 + 89}{4}

\frac{3 + 89}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 3 + 89}{4}

= \frac{2 3 + 89}{4} i

= \frac{2 3 + 89}{4} i

Simplify

- \frac{- 3 - 89}{8} : - i \frac{2 3 + 89}{4}

- \frac{- 3 - 89}{8}

Simplify

\frac{- 3 - 89}{8} : \frac{i 3 + 89}{2 2}

\frac{- 3 - 89}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{- 3 - 89}{8}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= \frac{- 3 - 89}{2 2}

Apply imaginary number rule:

- a = i a

= \frac{i 3 + 89}{2 2}

= - \frac{i 3 + 89}{2 2}

Rationalize

- \frac{i 3 + 89}{2 2} : - \frac{2 i 3 + 89}{4}

- \frac{i 3 + 89}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{i 3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 i 3 + 89}{4}

= - \frac{2 i 3 + 89}{4}

Rewrite

- \frac{2 i 3 + 89}{4}

in standard complex form:

- \frac{2 3 + 89}{4} i

- \frac{2 i 3 + 89}{4}

Cancel

\frac{2 i 3 + 89}{4} : \frac{i 3 + 89}{2 2}

\frac{2 i 3 + 89}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2 i 3 + 89}{2 ^{2}}

Cancel

\frac{2 i 3 + 89}{2 ^{2}} : \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

\frac{2 i 3 + 89}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} i 3 + 89}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{i 3 + 89}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

= \frac{i 3 + 89}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= \frac{i 3 + 89}{2 2}

= - \frac{i 3 + 89}{2 2}

- \frac{3 + 89}{2 2} = - \frac{2 3 + 89}{4}

- \frac{3 + 89}{2 2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{3 + 89 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= - \frac{2 3 + 89}{4}

= - \frac{2 3 + 89}{4} i

= - \frac{2 3 + 89}{4} i

u = i \frac{2 3 + 89}{4}, u = - i \frac{2 3 + 89}{4}

The solutions are

u = \frac{2 - 3 + 89}{4}, u = - \frac{2 - 3 + 89}{4}, u = i \frac{2 3 + 89}{4}, u = - i \frac{2 3 + 89}{4}

Substitute back

u = cos (x)

cos (x) = \frac{2 - 3 + 89}{4}, cos (x) = - \frac{2 - 3 + 89}{4}, cos (x) = i \frac{2 3 + 89}{4}, cos (x) = - i \frac{2 3 + 89}{4}

cos (x) = \frac{2 - 3 + 89}{4}, cos (x) = - \frac{2 - 3 + 89}{4}, cos (x) = i \frac{2 3 + 89}{4}, cos (x) = - i \frac{2 3 + 89}{4}

cos (x) = \frac{2 - 3 + 89}{4} : x = arccos (\frac{2 - 3 + 89}{4}) + 2 πn, x = 2 π - arccos (\frac{2 - 3 + 89}{4}) + 2 πn

cos (x) = \frac{2 - 3 + 89}{4}

Apply trig inverse properties

cos (x) = \frac{2 - 3 + 89}{4}

General solutions for

cos (x) = \frac{2 - 3 + 89}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2 - 3 + 89}{4}) + 2 πn, x = 2 π - arccos (\frac{2 - 3 + 89}{4}) + 2 πn

x = arccos (\frac{2 - 3 + 89}{4}) + 2 πn, x = 2 π - arccos (\frac{2 - 3 + 89}{4}) + 2 πn

cos (x) = - \frac{2 - 3 + 89}{4} : x = arccos (- \frac{2 - 3 + 89}{4}) + 2 πn, x = - arccos (- \frac{2 - 3 + 89}{4}) + 2 πn

cos (x) = - \frac{2 - 3 + 89}{4}

Apply trig inverse properties

cos (x) = - \frac{2 - 3 + 89}{4}

General solutions for

cos (x) = - \frac{2 - 3 + 89}{4}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{2 - 3 + 89}{4}) + 2 πn, x = - arccos (- \frac{2 - 3 + 89}{4}) + 2 πn

x = arccos (- \frac{2 - 3 + 89}{4}) + 2 πn, x = - arccos (- \frac{2 - 3 + 89}{4}) + 2 πn

cos (x) = i \frac{2 3 + 89}{4} :

No Solution

cos (x) = i \frac{2 3 + 89}{4}

No Solution

cos (x) = - i \frac{2 3 + 89}{4} :

No Solution

cos (x) = - i \frac{2 3 + 89}{4}

No Solution

Combine all the solutions

x = arccos (\frac{2 - 3 + 89}{4}) + 2 πn, x = 2 π - arccos (\frac{2 - 3 + 89}{4}) + 2 πn, x = arccos (- \frac{2 - 3 + 89}{4}) + 2 πn, x = - arccos (- \frac{2 - 3 + 89}{4}) + 2 πn

Show solutions in decimal form

x = 0.45831 \dots + 2 πn, x = 2 π - 0.45831 \dots + 2 πn, x = 2.68327 \dots + 2 πn, x = - 2.68327 \dots + 2 πn

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