All categories

3 years ago

(x+2/x-7) - (x^2+4x+13/x^2-4x-21)

Mathematics

2 answers:

olya-2409 [2.1K]3 years ago

7 0

Answer:

x = -2.98079 or x = -1.15272 or x = 0.892002 or x = 4.24151

Step-by-step explanation:

Solve for x:

-x^2 + x + 14 + 2/x - 13/x^2 = 0

Bring -x^2 + x + 14 + 2/x - 13/x^2 together using the common denominator x^2:

(-x^4 + x^3 + 14 x^2 + 2 x - 13)/x^2 = 0

Multiply both sides by x^2:

-x^4 + x^3 + 14 x^2 + 2 x - 13 = 0

Multiply both sides by -1:

x^4 - x^3 - 14 x^2 - 2 x + 13 = 0

Eliminate the cubic term by substituting y = x - 1/4:

13 - 2 (y + 1/4) - 14 (y + 1/4)^2 - (y + 1/4)^3 + (y + 1/4)^4 = 0

Expand out terms of the left hand side:

y^4 - (115 y^2)/8 - (73 y)/8 + 2973/256 = 0

Add (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8 to both sides:

y^4 + (sqrt(2973) y^2)/8 + 2973/256 = (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8

y^4 + (sqrt(2973) y^2)/8 + 2973/256 = (y^2 + sqrt(2973)/16)^2:

(y^2 + sqrt(2973)/16)^2 = (sqrt(2973) y^2)/8 + (115 y^2)/8 + (73 y)/8

Add 2 (y^2 + sqrt(2973)/16) λ + λ^2 to both sides:

(y^2 + sqrt(2973)/16)^2 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2

(y^2 + sqrt(2973)/16)^2 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (y^2 + sqrt(2973)/16 + λ)^2:

(y^2 + sqrt(2973)/16 + λ)^2 = (73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2

(73 y)/8 + (sqrt(2973) y^2)/8 + (115 y^2)/8 + 2 λ (y^2 + sqrt(2973)/16) + λ^2 = (2 λ + 115/8 + sqrt(2973)/8) y^2 + (73 y)/8 + (sqrt(2973) λ)/8 + λ^2:

(y^2 + sqrt(2973)/16 + λ)^2 = y^2 (2 λ + 115/8 + sqrt(2973)/8) + (73 y)/8 + (sqrt(2973) λ)/8 + λ^2

Complete the square on the right hand side:

(y^2 + sqrt(2973)/16 + λ)^2 = (y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)))^2 + (4 (2 λ + 115/8 + sqrt(2973)/8) (λ^2 + (sqrt(2973) λ)/8) - 5329/64)/(4 (2 λ + 115/8 + sqrt(2973)/8))

To express the right hand side as a square, find a value of λ such that the last term is 0.

This means 4 (2 λ + 115/8 + sqrt(2973)/8) (λ^2 + (sqrt(2973) λ)/8) - 5329/64 = 1/64 (512 λ^3 + 96 sqrt(2973) λ^2 + 3680 λ^2 + 460 sqrt(2973) λ + 11892 λ - 5329) = 0.

Thus the root λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3)) allows the right hand side to be expressed as a square.

(This value will be substituted later):

(y^2 + sqrt(2973)/16 + λ)^2 = (y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)))^2

Take the square root of both sides:

y^2 + sqrt(2973)/16 + λ = y sqrt(2 λ + 115/8 + sqrt(2973)/8) + 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8)) or y^2 + sqrt(2973)/16 + λ = -y sqrt(2 λ + 115/8 + sqrt(2973)/8) - 73/(16 sqrt(2 λ + 115/8 + sqrt(2973)/8))

Solve using the quadratic formula:

y = 1/8 (sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) + sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 + 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) or y = 1/8 (sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) - sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 + 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) or y = 1/8 (sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 - 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973))) - sqrt(2) sqrt(16 λ + 115 + sqrt(2973))) or y = 1/8 (-sqrt(2) sqrt(16 λ + 115 + sqrt(2973)) - sqrt(2) sqrt((10252 - 32 sqrt(2973) λ - 256 λ^2 - 292 sqrt(2) sqrt(16 λ + 115 + sqrt(2973)))/(16 λ + 115 + sqrt(2973)))) where λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3))

Substitute λ = 1/48 (-3 sqrt(2973) - 115) + 1/12 (-i sqrt(3) + 1) ((3 i sqrt(10705335) - 8327)/2)^(1/3) + (173 (i sqrt(3) + 1))/(3 2^(2/3) (3 i sqrt(10705335) - 8327)^(1/3)) and approximate:

y = -3.23079 or y = -1.40272 or y = 0.642002 or y = 3.99151

Substitute back for y = x - 1/4:

x - 1/4 = -3.23079 or y = -1.40272 or y = 0.642002 or y = 3.99151

Add 1/4 to both sides:

x = -2.98079 or y = -1.40272 or y = 0.642002 or y = 3.99151

Substitute back for y = x - 1/4:

x = -2.98079 or x - 1/4 = -1.40272 or y = 0.642002 or y = 3.99151

Add 1/4 to both sides:

x = -2.98079 or x = -1.15272 or y = 0.642002 or y = 3.99151

Substitute back for y = x - 1/4:

x = -2.98079 or x = -1.15272 or x - 1/4 = 0.642002 or y = 3.99151

Add 1/4 to both sides:

x = -2.98079 or x = -1.15272 or x = 0.892002 or y = 3.99151

Substitute back for y = x - 1/4:

x = -2.98079 or x = -1.15272 or x = 0.892002 or x - 1/4 = 3.99151

Add 1/4 to both sides:

Answer: x = -2.98079 or x = -1.15272 or x = 0.892002 or x = 4.24151

allochka39001 [22]3 years ago

6 0

Step-by-step explanation:

$( \frac{x + 2}{x - 7}) - ( \frac{ {x}^{2} + 4x + 13 }{ {x}^{2} - 4x - 21} ) \\ \\ = ( \frac{x + 2}{x - 7}) - ( \frac{ {x}^{2} + 4x + 13 }{ {x}^{2} - 7x + 3x- 21} )\\ \\ = ( \frac{x + 2}{x - 7}) - ( \frac{ {x}^{2} + 4x + 13 }{x( {x} - 7) + 3(x- 7)} )\\ \\ = ( \frac{x + 2}{x - 7}) - ( \frac{ {x}^{2} + 4x + 13 }{( {x} - 7) (x + 3)} )\\ \\ = ( \frac{(x + 2)(x + 3)}{(x - 7)(x + 3)}) - ( \frac{ {x}^{2} + 4x + 13 }{( {x} - 7) (x + 3)} ) \\ \\ = \frac{x^{2} + 5x + 6}{(x - 7)(x + 3)} - \frac{ {x}^{2} + 4x + 13 }{( {x} - 7) (x + 3)} \\ \\ = \frac{x^{2} + 5x + 6 - ({x}^{2} + 4x + 13 \: )}{(x - 7)(x + 3)} \\ \\ = \frac{x^{2} + 5x + 6 - {x}^{2} - 4x - 13 \: }{(x - 7)(x + 3)} \\ \\ = \frac{x^{2} + 5x + 6 - {x}^{2} - 4x - 13 \: }{(x - 7)(x + 3)} \\ \\ = \frac{ x - 7\: }{(x - 7)(x + 3)} \\ \\ = \frac{ 1}{(x + 3)} \\ \\ \purple{ \boxed{\therefore( \frac{x + 2}{x - 7}) - ( \frac{ {x}^{2} + 4x + 13 }{ {x}^{2} - 4x - 21} ) = \frac{ 1}{(x + 3)}}} \\ \\$

You might be interested in

Answer plzzz I’ll give u more points

Olenka [21]

13. $9 because 75%=3/4 and 1/4 of 12 is 3, and so you multiply $3 by 3 to get $9
14. Multiply 325 by .2 to get $65.00
15. 15 minutes, because if it rises 2 feet every hour, then it rises 1 foot every hour, and so you divide that by 2 to get 15
Answers:
13. $9
14. $65.00
15. 15 mins

4 0

3 years ago

Mika wants to buy a condominium. He has the choice of buying it now or renting it with the option to buy at the end of 3 years.

mixer [17]

Answer:

Step-by-step explanation:

$828

5 0

3 years ago

Read 2 more answers

It shows the equations but it is a little confusing I don't want to get the problem wrong please help me out. Ill give you brain

Viktor [21]

Answer:εδΑΒΓΒΕ

$U4\leq \sqrt{x} \sqrt[n]{x} 64$

Step-by-step explanation:

73hsay is a little bit too long and ∩679∨78ω8㏒∴≠÷±

6 0

3 years ago

Jasmine's plane ticket cost $466.88. She has already saved 25% of the cost of the ticket. During the next 2 weeks, she will save

tresset_1 [31]

Answer: She would save $175.08 during each of the next 2 weeks.

Step-by-step explanation:

Jasmine's plane ticket cost $466.88. She has already saved 25% of the cost of the ticket. This means that the amount that she already saved is

25/100 × 466.88 = $116.72

The amount remaining would be

466.88 - 116.72 = $350.16

During the next 2 weeks, she will save 1/2 of the remaining cost of the ticket. This means that the amount that she would save during each of the next 2 weeks is

1/2 × 350.16 = $175.08

3 0

3 years ago

Mrs. Carlson wrote the following question for her class: "Ricardo's basketball team is selling raffle tickets, Each ticket costs

IceJOKER [234]

Answer:

michelle, kaley, and Rebecca.

Step-by-step explanation:

They have sold 100 tickets so far ($75), they need to sell 380 ($285) tickets in total and need to sell 280 more tickets.

Michelle was wrong because she should have divided 210 (how much more money they need to sell on tickets) into .75 (which is how much each ticket costs).

Kaley was wrong because she should have also divided 210 into .75 like Michelle.

Rebecca should have also divided 210 into .75.

Eva was the only correct one.

7 0

4 years ago