What is the general solution for arctan(x+1x-1)+arctan(x-12)=arctan(2) ?

The general solution for arctan(x+1x-1)+arctan(x-12)=arctan(2) is x=(47+sqrt(2065))/8

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arctan(x+1x-1)+arctan(x-12)=arctan(2)

Solution

arctan (x + 1 x - 1) + arctan (x - 12) = arctan (2)

Solution

x = \frac{47 + 2065}{8}

Solution steps

arctan (x + 1 \cdot x - 1) + arctan (x - 12) = arctan (2)

Rewrite using trig identities

arctan (x + 1 \cdot x - 1) + arctan (x - 12)

Use the Sum to Product identity:

arctan (s) + arctan (t) = arctan (\frac{s + t}{1 - s t})

= arctan (\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )})

arctan (\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )}) = arctan (2)

Apply trig inverse properties

arctan (\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )}) = arctan (2)

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

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} = tan (arctan (2))

tan (arctan (2)) = 2

tan (arctan (2))

Rewrite using trig identities:

tan (arctan (2)) = 2

Use the following identity:

tan (arctan (x)) = x

= 2

= 2

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} = 2

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} = 2

Solve

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} = 2 : x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} = 2

Simplify

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )} : \frac{3 x - 13}{- 2 x ^{2} + 25 x - 11}

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )}

x + 1 \cdot x - 1 + x - 12 = 3 x - 13

x + 1 \cdot x - 1 + x - 12

Group like terms

= x + 1 \cdot x + x - 1 - 12

Add similar elements:

x + 1 \cdot x + x = 3 x

= 3 x - 1 - 12

Subtract the numbers:

- 1 - 12 = - 13

= 3 x - 13

= \frac{3 x - 13}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )}

Add similar elements:

x + 1 \cdot x = 2 x

= \frac{3 x - 13}{1 - ( 2 x - 1 ) ( x - 12 )}

Expand

1 - (2 x - 1) (x - 12) : - 2 x^{2} + 25 x - 11

1 - (2 x - 1) (x - 12)

Expand

- (2 x - 1) (x - 12) : - 2 x^{2} + 25 x - 12

Expand

(2 x - 1) (x - 12) : 2 x^{2} - 25 x + 12

(2 x - 1) (x - 12)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 2 x, b = - 1, c = x, d = - 12

= 2 xx + 2 x (- 12) + (- 1) x + (- 1) (- 12)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12

Simplify

2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12 : 2 x^{2} - 25 x + 12

2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12

2 xx = 2 x^{2}

2 xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 2 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 x^{2}

2 \cdot 12 x = 24 x

2 \cdot 12 x

Multiply the numbers:

2 \cdot 12 = 24

= 24 x

1 \cdot x = x

1 \cdot x

Multiply:

1 \cdot x = x

= x

1 \cdot 12 = 12

1 \cdot 12

Multiply the numbers:

1 \cdot 12 = 12

= 12

= 2 x^{2} - 24 x - x + 12

Add similar elements:

- 24 x - x = - 25 x

= 2 x^{2} - 25 x + 12

= 2 x^{2} - 25 x + 12

= - (2 x^{2} - 25 x + 12)

Distribute parentheses

= - (2 x^{2}) - (- 25 x) - (12)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 x^{2} + 25 x - 12

= 1 - 2 x^{2} + 25 x - 12

Simplify

1 - 2 x^{2} + 25 x - 12 : - 2 x^{2} + 25 x - 11

1 - 2 x^{2} + 25 x - 12

Group like terms

= - 2 x^{2} + 25 x + 1 - 12

Add/Subtract the numbers:

1 - 12 = - 11

= - 2 x^{2} + 25 x - 11

= - 2 x^{2} + 25 x - 11

= \frac{3 x - 13}{- 2 x ^{2} + 25 x - 11}

\frac{3 x - 13}{- 2 x ^{2} + 25 x - 11} = 2

Multiply both sides by

- 2 x^{2} + 25 x - 11

\frac{3 x - 13}{- 2 x ^{2} + 25 x - 11} = 2

Multiply both sides by

- 2 x^{2} + 25 x - 11

\frac{3 x - 13}{- 2 x ^{2} + 25 x - 11} (- 2 x^{2} + 25 x - 11) = 2 (- 2 x^{2} + 25 x - 11)

Simplify

3 x - 13 = 2 (- 2 x^{2} + 25 x - 11)

3 x - 13 = 2 (- 2 x^{2} + 25 x - 11)

Solve

3 x - 13 = 2 (- 2 x^{2} + 25 x - 11) : x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

3 x - 13 = 2 (- 2 x^{2} + 25 x - 11)

Expand

2 (- 2 x^{2} + 25 x - 11) : - 4 x^{2} + 50 x - 22

2 (- 2 x^{2} + 25 x - 11)

Distribute parentheses

= 2 (- 2 x^{2}) + 2 \cdot 25 x + 2 (- 11)

Apply minus-plus rules

+ (- a) = - a

= - 2 \cdot 2 x^{2} + 2 \cdot 25 x - 2 \cdot 11

Simplify

- 2 \cdot 2 x^{2} + 2 \cdot 25 x - 2 \cdot 11 : - 4 x^{2} + 50 x - 22

- 2 \cdot 2 x^{2} + 2 \cdot 25 x - 2 \cdot 11

Multiply the numbers:

2 \cdot 2 = 4

= - 4 x^{2} + 2 \cdot 25 x - 2 \cdot 11

Multiply the numbers:

2 \cdot 25 = 50

= - 4 x^{2} + 50 x - 2 \cdot 11

Multiply the numbers:

2 \cdot 11 = 22

= - 4 x^{2} + 50 x - 22

= - 4 x^{2} + 50 x - 22

3 x - 13 = - 4 x^{2} + 50 x - 22

Switch sides

- 4 x^{2} + 50 x - 22 = 3 x - 13

Move

13

to the left side

- 4 x^{2} + 50 x - 22 = 3 x - 13

Add

13

to both sides

- 4 x^{2} + 50 x - 22 + 13 = 3 x - 13 + 13

Simplify

- 4 x^{2} + 50 x - 9 = 3 x

- 4 x^{2} + 50 x - 9 = 3 x

Move

3 x

to the left side

- 4 x^{2} + 50 x - 9 = 3 x

Subtract

3 x

from both sides

- 4 x^{2} + 50 x - 9 - 3 x = 3 x - 3 x

Simplify

- 4 x^{2} + 47 x - 9 = 0

- 4 x^{2} + 47 x - 9 = 0

Solve with the quadratic formula

- 4 x^{2} + 47 x - 9 = 0

Quadratic Equation Formula:

For

a = - 4, b = 47, c = - 9

x_{1, 2} = \frac{- 47 \pm 4 7 ^{2} - 4 ( - 4 ) ( - 9 )}{2 ( - 4 )}

x_{1, 2} = \frac{- 47 \pm 4 7 ^{2} - 4 ( - 4 ) ( - 9 )}{2 ( - 4 )}

4 7^{2} - 4 (- 4) (- 9) = 2065

4 7^{2} - 4 (- 4) (- 9)

Apply rule

- (- a) = a

= 4 7^{2} - 4 \cdot 4 \cdot 9

Multiply the numbers:

4 \cdot 4 \cdot 9 = 144

= 4 7^{2} - 144

4 7^{2} = 2209

= 2209 - 144

Subtract the numbers:

2209 - 144 = 2065

= 2065

x_{1, 2} = \frac{- 47 \pm 2065}{2 ( - 4 )}

Separate the solutions

x_{1} = \frac{- 47 + 2065}{2 ( - 4 )}, x_{2} = \frac{- 47 - 2065}{2 ( - 4 )}

x = \frac{- 47 + 2065}{2 ( - 4 )} : \frac{47 - 2065}{8}

\frac{- 47 + 2065}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 47 + 2065}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 47 + 2065}{- 8}

Apply the fraction rule:

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

- 47 + 2065 = - (47 - 2065)

= \frac{47 - 2065}{8}

x = \frac{- 47 - 2065}{2 ( - 4 )} : \frac{47 + 2065}{8}

\frac{- 47 - 2065}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 47 - 2065}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 47 - 2065}{- 8}

Apply the fraction rule:

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

- 47 - 2065 = - (47 + 2065)

= \frac{47 + 2065}{8}

The solutions to the quadratic equation are:

x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

Verify Solutions

Find undefined (singularity) points:

x = \frac{25 - 537}{4}, x = \frac{25 + 537}{4}

Take the denominator(s) of

\frac{x + 1 \cdot x - 1 + x - 12}{1 - ( x + 1 \cdot x - 1 ) ( x - 12 )}

and compare to zero

Solve

1 - (x + 1 \cdot x - 1) (x - 12) = 0 : x = \frac{25 - 537}{4}, x = \frac{25 + 537}{4}

1 - (x + 1 \cdot x - 1) (x - 12) = 0

Expand

1 - (x + 1 \cdot x - 1) (x - 12) : - 2 x^{2} + 25 x - 11

1 - (x + 1 \cdot x - 1) (x - 12)

Add similar elements:

x + 1 \cdot x = 2 x

= 1 - (2 x - 1) (x - 12)

Expand

- (2 x - 1) (x - 12) : - 2 x^{2} + 25 x - 12

Expand

(2 x - 1) (x - 12) : 2 x^{2} - 25 x + 12

(2 x - 1) (x - 12)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = 2 x, b = - 1, c = x, d = - 12

= 2 xx + 2 x (- 12) + (- 1) x + (- 1) (- 12)

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

= 2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12

Simplify

2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12 : 2 x^{2} - 25 x + 12

2 xx - 2 \cdot 12 x - 1 \cdot x + 1 \cdot 12

2 xx = 2 x^{2}

2 xx

Apply exponent rule:

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

xx = x^{1 + 1}

= 2 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 x^{2}

2 \cdot 12 x = 24 x

2 \cdot 12 x

Multiply the numbers:

2 \cdot 12 = 24

= 24 x

1 \cdot x = x

1 \cdot x

Multiply:

1 \cdot x = x

= x

1 \cdot 12 = 12

1 \cdot 12

Multiply the numbers:

1 \cdot 12 = 12

= 12

= 2 x^{2} - 24 x - x + 12

Add similar elements:

- 24 x - x = - 25 x

= 2 x^{2} - 25 x + 12

= 2 x^{2} - 25 x + 12

= - (2 x^{2} - 25 x + 12)

Distribute parentheses

= - (2 x^{2}) - (- 25 x) - (12)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 x^{2} + 25 x - 12

= 1 - 2 x^{2} + 25 x - 12

Simplify

1 - 2 x^{2} + 25 x - 12 : - 2 x^{2} + 25 x - 11

1 - 2 x^{2} + 25 x - 12

Group like terms

= - 2 x^{2} + 25 x + 1 - 12

Add/Subtract the numbers:

1 - 12 = - 11

= - 2 x^{2} + 25 x - 11

= - 2 x^{2} + 25 x - 11

- 2 x^{2} + 25 x - 11 = 0

Solve with the quadratic formula

- 2 x^{2} + 25 x - 11 = 0

Quadratic Equation Formula:

For

a = - 2, b = 25, c = - 11

x_{1, 2} = \frac{- 25 \pm 2 5 ^{2} - 4 ( - 2 ) ( - 11 )}{2 ( - 2 )}

x_{1, 2} = \frac{- 25 \pm 2 5 ^{2} - 4 ( - 2 ) ( - 11 )}{2 ( - 2 )}

2 5^{2} - 4 (- 2) (- 11) = 537

2 5^{2} - 4 (- 2) (- 11)

Apply rule

- (- a) = a

= 2 5^{2} - 4 \cdot 2 \cdot 11

Multiply the numbers:

4 \cdot 2 \cdot 11 = 88

= 2 5^{2} - 88

2 5^{2} = 625

= 625 - 88

Subtract the numbers:

625 - 88 = 537

= 537

x_{1, 2} = \frac{- 25 \pm 537}{2 ( - 2 )}

Separate the solutions

x_{1} = \frac{- 25 + 537}{2 ( - 2 )}, x_{2} = \frac{- 25 - 537}{2 ( - 2 )}

x = \frac{- 25 + 537}{2 ( - 2 )} : \frac{25 - 537}{4}

\frac{- 25 + 537}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 25 + 537}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 25 + 537}{- 4}

Apply the fraction rule:

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

- 25 + 537 = - (25 - 537)

= \frac{25 - 537}{4}

x = \frac{- 25 - 537}{2 ( - 2 )} : \frac{25 + 537}{4}

\frac{- 25 - 537}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 25 - 537}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 25 - 537}{- 4}

Apply the fraction rule:

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

- 25 - 537 = - (25 + 537)

= \frac{25 + 537}{4}

The solutions to the quadratic equation are:

x = \frac{25 - 537}{4}, x = \frac{25 + 537}{4}

The following points are undefined

x = \frac{25 - 537}{4}, x = \frac{25 + 537}{4}

Combine undefined points with solutions:

x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

x = \frac{47 - 2065}{8}, x = \frac{47 + 2065}{8}

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (x + 1 x - 1) + arctan (x - 12) = arctan (2)

Remove the ones that don't agree with the equation.

Check the solution

\frac{47 - 2065}{8} :

False

\frac{47 - 2065}{8}

Plug in

n = 1

\frac{47 - 2065}{8}

For

arctan (x + 1 x - 1) + arctan (x - 12) = arctan (2)

plug in

x = \frac{47 - 2065}{8}

arctan (\frac{47 - 2065}{8} + 1 \cdot \frac{47 - 2065}{8} - 1) + arctan (\frac{47 - 2065}{8} - 12) = arctan (2)

Refine

- 2.03444 \dots = 1.10714 \dots

\Rightarrow False

Check the solution

\frac{47 + 2065}{8} :

True

\frac{47 + 2065}{8}

Plug in

n = 1

\frac{47 + 2065}{8}

For

arctan (x + 1 x - 1) + arctan (x - 12) = arctan (2)

plug in

x = \frac{47 + 2065}{8}

arctan (\frac{47 + 2065}{8} + 1 \cdot \frac{47 + 2065}{8} - 1) + arctan (\frac{47 + 2065}{8} - 12) = arctan (2)

Refine

1.10714 \dots = 1.10714 \dots

\Rightarrow True

x = \frac{47 + 2065}{8}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for arctan(x+1x-1)+arctan(x-12)=arctan(2) ?
The general solution for arctan(x+1x-1)+arctan(x-12)=arctan(2) is x=(47+sqrt(2065))/8