What is the general solution for 4tan^2(x)+7tan(x)-4=0 ?

The general solution for 4tan^2(x)+7tan(x)-4=0 is x=0.42598…+pin,x=-1.14481…+pin

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4tan^2(x)+7tan(x)-4=0

4 tan^{2} (x) + 7 tan (x) - 4 = 0

x = 0.42598 \dots + πn, x = - 1.14481 \dots + πn

Degrees

x = 24.40703 \dots^{\circ} + 18 0^{\circ} n, x = - 65.59296 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 tan^{2} (x) + 7 tan (x) - 4 = 0

Solve by substitution

4 tan^{2} (x) + 7 tan (x) - 4 = 0

Let:

tan (x) = u

4 u^{2} + 7 u - 4 = 0

4 u^{2} + 7 u - 4 = 0 : u = \frac{- 7 + 113}{8}, u = \frac{- 7 - 113}{8}

4 u^{2} + 7 u - 4 = 0

Solve with the quadratic formula

4 u^{2} + 7 u - 4 = 0

Quadratic Equation Formula:

For

a = 4, b = 7, c = - 4

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 4 ( - 4 )}{2 \cdot 4}

7^{2} - 4 \cdot 4 (- 4) = 113

7^{2} - 4 \cdot 4 (- 4)

Apply rule

- (- a) = a

= 7^{2} + 4 \cdot 4 \cdot 4

Multiply the numbers:

4 \cdot 4 \cdot 4 = 64

= 7^{2} + 64

7^{2} = 49

= 49 + 64

Add the numbers:

49 + 64 = 113

= 113

u_{1, 2} = \frac{- 7 \pm 113}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- 7 + 113}{2 \cdot 4}, u_{2} = \frac{- 7 - 113}{2 \cdot 4}

u = \frac{- 7 + 113}{2 \cdot 4} : \frac{- 7 + 113}{8}

\frac{- 7 + 113}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 7 + 113}{8}

u = \frac{- 7 - 113}{2 \cdot 4} : \frac{- 7 - 113}{8}

\frac{- 7 - 113}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 7 - 113}{8}

The solutions to the quadratic equation are:

u = \frac{- 7 + 113}{8}, u = \frac{- 7 - 113}{8}

Substitute back

u = tan (x)

tan (x) = \frac{- 7 + 113}{8}, tan (x) = \frac{- 7 - 113}{8}

tan (x) = \frac{- 7 + 113}{8}, tan (x) = \frac{- 7 - 113}{8}

tan (x) = \frac{- 7 + 113}{8} : x = arctan (\frac{- 7 + 113}{8}) + πn

tan (x) = \frac{- 7 + 113}{8}

Apply trig inverse properties

tan (x) = \frac{- 7 + 113}{8}

General solutions for

tan (x) = \frac{- 7 + 113}{8}

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

x = arctan (\frac{- 7 + 113}{8}) + πn

x = arctan (\frac{- 7 + 113}{8}) + πn

tan (x) = \frac{- 7 - 113}{8} : x = arctan (\frac{- 7 - 113}{8}) + πn

tan (x) = \frac{- 7 - 113}{8}

Apply trig inverse properties

tan (x) = \frac{- 7 - 113}{8}

General solutions for

tan (x) = \frac{- 7 - 113}{8}

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

x = arctan (\frac{- 7 - 113}{8}) + πn

x = arctan (\frac{- 7 - 113}{8}) + πn

Combine all the solutions

x = arctan (\frac{- 7 + 113}{8}) + πn, x = arctan (\frac{- 7 - 113}{8}) + πn

Show solutions in decimal form

x = 0.42598 \dots + πn, x = - 1.14481 \dots + πn

What is the general solution for 4tan^2(x)+7tan(x)-4=0 ?
The general solution for 4tan^2(x)+7tan(x)-4=0 is x=0.42598…+pin,x=-1.14481…+pin