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solvefor x,arctan(x^2+9y^2-2x-36y+37)=0

Solution

solve for

x, arctan (x^{2} + 9 y^{2} - 2 x - 36 y + 37) = 0

Solution

x = 1 + 3 - y^{2} + 4 y - 4, x = 1 - 3 - y^{2} + 4 y - 4

Solution steps

arctan (x^{2} + 9 y^{2} - 2 x - 36 y + 37) = 0

Apply trig inverse properties

arctan (x^{2} + 9 y^{2} - 2 x - 36 y + 37) = 0

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

x^{2} + 9 y^{2} - 2 x - 36 y + 37 = tan (0)

tan (0) = 0

tan (0)

Use the following trivial identity:

tan (0) = 0

tan (0)

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

x^{2} + 9 y^{2} - 2 x - 36 y + 37 = 0

x^{2} + 9 y^{2} - 2 x - 36 y + 37 = 0

Solve

x^{2} + 9 y^{2} - 2 x - 36 y + 37 = 0 : x = 1 + 3 - y^{2} + 4 y - 4, x = 1 - 3 - y^{2} + 4 y - 4

x^{2} + 9 y^{2} - 2 x - 36 y + 37 = 0

Write in the standard form

a x^{2} + b x + c = 0

x^{2} - 2 x + 9 y^{2} - 36 y + 37 = 0

Solve with the quadratic formula

x^{2} - 2 x + 9 y^{2} - 36 y + 37 = 0

Quadratic Equation Formula:

For

a = 1, b = - 2, c = 9 y^{2} - 36 y + 37

x_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( 9 y ^{2} - 36 y + 37 )}{2 \cdot 1}

x_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( 9 y ^{2} - 36 y + 37 )}{2 \cdot 1}

Simplify

(- 2)^{2} - 4 \cdot 1 \cdot (9 y^{2} - 36 y + 37) : 6 - y^{2} + 4 y - 4

(- 2)^{2} - 4 \cdot 1 \cdot (9 y^{2} - 36 y + 37)

Apply exponent rule:

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

n

is even

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

= 2^{2} - 4 \cdot 1 \cdot (9 y^{2} - 36 y + 37)

Multiply the numbers:

4 \cdot 1 = 4

= 2^{2} - 4 (9 y^{2} - 36 y + 37)

Factor

2^{2} - 4 (9 y^{2} - 36 y + 37) : 36 (- y^{2} + 4 y - 4)

2^{2} - 4 (9 y^{2} - 36 y + 37)

Rewrite as

= 4 \cdot 1 - 4 (37 + y^{2} \cdot 9 - 36 y)

Factor out common term

4

= 4 (1 - (37 + y^{2} \cdot 9 - 36 y))

Factor

- (9 y^{2} - 36 y + 37) + 1 : 9 (- y^{2} + 4 y - 4)

1 - (37 + y^{2} \cdot 9 - 36 y)

= 1 - (37 + 9 y^{2} - 36 y)

- (37 + y^{2} \cdot 9 - 36 y) : - 37 - y^{2} \cdot 9 + 36 y

- (37 + y^{2} \cdot 9 - 36 y)

Distribute parentheses

= - 37 - y^{2} \cdot 9 - (- 36 y)

Apply minus-plus rules

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

= - 37 - y^{2} \cdot 9 + 36 y

= 1 - 37 - y^{2} \cdot 9 + 36 y

Subtract the numbers:

1 - 37 = - 36

= - 9 y^{2} + 36 y - 36

Rewrite as

= - 9 y^{2} + 9 \cdot 4 y - 9 \cdot 4

Factor out common term

9

= 9 (- y^{2} + 4 y - 4)

= 4 \cdot 9 (- y^{2} + 4 y - 4)

Refine

= 36 (- y^{2} + 4 y - 4)

= 36 (- y^{2} + 4 y - 4)

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 36 - y^{2} + 4 y - 4

36 = 6

36

Factor the number:

36 = 6^{2}

= 6^{2}

Apply radical rule:

6^{2} = 6

= 6

= 6 - y^{2} + 4 y - 4

Factor

- y^{2} + 4 y - 4 : - (y - 2)^{2}

- y^{2} + 4 y - 4

Factor out common term

- 1

= - (y^{2} - 4 y + 4)

Factor

y^{2} - 4 y + 4 : (y - 2) (y - 2)

y^{2} - 4 y + 4

Break the expression into groups

y^{2} - 4 y + 4

Definition

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

The factors of

4

1, 2, 4

Negative factors of

4 : - 1, - 2, - 4

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 4

For every two factors such that

u * v = 4,

check if

u + v = - 4

Check

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

FalseCheck

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

False

u = - 2, v = - 2

Group into

(a x^{2} + ux) + (vx + c)

y^{2} - 2 y - 2 y + 4

= y^{2} - 2 y - 2 y + 4

Factor out

y

from

y^{2} - 2 y : y (y - 2)

y^{2} - 2 y

Apply exponent rule:

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

y^{2} = yy

= yy - 2 y

Factor out common term

y

= y (y - 2)

Factor out

- 2

from

- 2 y + 4 : - 2 (y - 2)

- 2 y + 4

Rewrite

4

2 \cdot 2

= - 2 y + 2 \cdot 2

Factor out common term

- 2

= - 2 (y - 2)

= y (y - 2) - 2 (y - 2)

Factor out common term

y - 2

= (y - 2) (y - 2)

= - (y - 2) (y - 2)

Refine

= - (y - 2)^{2}

= 6 - (y - 2)^{2}

- (y - 2)^{2} = - y^{2} + 4 y - 4

- (y - 2)^{2}

Expand

- (y - 2)^{2} : - y^{2} + 4 y - 4

- (y - 2)^{2}

(y - 2)^{2} : y^{2} - 4 y + 4

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = y, b = 2

= y^{2} - 2 y \cdot 2 + 2^{2}

Simplify

y^{2} - 2 y \cdot 2 + 2^{2} : y^{2} - 4 y + 4

y^{2} - 2 y \cdot 2 + 2^{2}

Multiply the numbers:

2 \cdot 2 = 4

= y^{2} - 4 y + 2^{2}

2^{2} = 4

= y^{2} - 4 y + 4

= y^{2} - 4 y + 4

= - (y^{2} - 4 y + 4)

Expand

- (y^{2} - 4 y + 4) : - y^{2} + 4 y - 4

Distribute parentheses

= - y^{2} - (- 4 y) - 4

Apply minus-plus rules

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

= - y^{2} + 4 y - 4

= - y^{2} + 4 y - 4

= - y^{2} + 4 y - 4

= 6 - y^{2} + 4 y - 4

x_{1, 2} = \frac{- ( - 2 ) \pm 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

Separate the solutions

x_{1} = \frac{- ( - 2 ) + 6 - y ^{2} + 4 y - 4}{2 \cdot 1}, x_{2} = \frac{- ( - 2 ) - 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

x = \frac{- ( - 2 ) + 6 - y ^{2} + 4 y - 4}{2 \cdot 1} : 1 + 3 - y^{2} + 4 y - 4

\frac{- ( - 2 ) + 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 + 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 6 - y ^{2} + 4 y - 4}{2}

Factor

2 + 6 - y^{2} + 4 y - 4 : 2 (1 + 3 - y^{2} - 4 + 4 y)

2 + 6 - y^{2} + 4 y - 4

Rewrite as

= 2 \cdot 1 + 2 \cdot 3 - y^{2} - 4 + 4 y

Factor out common term

2

= 2 (1 + 3 - y^{2} - 4 + 4 y)

= \frac{2 ( 1 + 3 - y ^{2} - 4 + 4 y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 3 - y^{2} + 4 y - 4

x = \frac{- ( - 2 ) - 6 - y ^{2} + 4 y - 4}{2 \cdot 1} : 1 - 3 - y^{2} + 4 y - 4

\frac{- ( - 2 ) - 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{2 - 6 - y ^{2} + 4 y - 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 6 - y ^{2} + 4 y - 4}{2}

Factor

2 - 6 - y^{2} + 4 y - 4 : 2 (1 - 3 - y^{2} - 4 + 4 y)

2 - 6 - y^{2} + 4 y - 4

Rewrite as

= 2 \cdot 1 - 2 \cdot 3 - y^{2} - 4 + 4 y

Factor out common term

2

= 2 (1 - 3 - y^{2} - 4 + 4 y)

= \frac{2 ( 1 - 3 - y ^{2} - 4 + 4 y )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 - 3 - y^{2} + 4 y - 4

The solutions to the quadratic equation are:

x = 1 + 3 - y^{2} + 4 y - 4, x = 1 - 3 - y^{2} + 4 y - 4

x = 1 + 3 - y^{2} + 4 y - 4, x = 1 - 3 - y^{2} + 4 y - 4

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