What is the general solution for 4tan^2(x)-8tan(x)+3=0 ?

The general solution for 4tan^2(x)-8tan(x)+3=0 is x=0.98279…+pin,x=0.46364…+pin

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4tan^2(x)-8tan(x)+3=0

Solution

4 tan^{2} (x) - 8 tan (x) + 3 = 0

Solution

x = 0.98279 \dots + πn, x = 0.46364 \dots + πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) - 8 tan (x) + 3 = 0

Solve by substitution

4 tan^{2} (x) - 8 tan (x) + 3 = 0

Let:

tan (x) = u

4 u^{2} - 8 u + 3 = 0

4 u^{2} - 8 u + 3 = 0 : u = \frac{3}{2}, u = \frac{1}{2}

4 u^{2} - 8 u + 3 = 0

Solve with the quadratic formula

4 u^{2} - 8 u + 3 = 0

Quadratic Equation Formula:

For

a = 4, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

(- 8)^{2} - 4 \cdot 4 \cdot 3 = 4

(- 8)^{2} - 4 \cdot 4 \cdot 3

Apply exponent rule:

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

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 4 \cdot 3

Multiply the numbers:

4 \cdot 4 \cdot 3 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

Subtract the numbers:

64 - 48 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 8 ) \pm 4}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 4}{2 \cdot 4}, u_{2} = \frac{- ( - 8 ) - 4}{2 \cdot 4}

u = \frac{- ( - 8 ) + 4}{2 \cdot 4} : \frac{3}{2}

\frac{- ( - 8 ) + 4}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{8 + 4}{2 \cdot 4}

Add the numbers:

8 + 4 = 12

= \frac{12}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{12}{8}

Cancel the common factor:

4

= \frac{3}{2}

u = \frac{- ( - 8 ) - 4}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 8 ) - 4}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{8 - 4}{2 \cdot 4}

Subtract the numbers:

8 - 4 = 4

= \frac{4}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = \frac{1}{2}

Substitute back

u = tan (x)

tan (x) = \frac{3}{2}, tan (x) = \frac{1}{2}

tan (x) = \frac{3}{2}, tan (x) = \frac{1}{2}

tan (x) = \frac{3}{2} : x = arctan (\frac{3}{2}) + πn

tan (x) = \frac{3}{2}

Apply trig inverse properties

tan (x) = \frac{3}{2}

General solutions for

tan (x) = \frac{3}{2}

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

tan (x) = \frac{1}{2} : x = arctan (\frac{1}{2}) + πn

tan (x) = \frac{1}{2}

Apply trig inverse properties

tan (x) = \frac{1}{2}

General solutions for

tan (x) = \frac{1}{2}

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

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

Combine all the solutions

x = arctan (\frac{3}{2}) + πn, x = arctan (\frac{1}{2}) + πn

Show solutions in decimal form

x = 0.98279 \dots + πn, x = 0.46364 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 4tan^2(x)-8tan(x)+3=0 ?
The general solution for 4tan^2(x)-8tan(x)+3=0 is x=0.98279…+pin,x=0.46364…+pin