Question

(4х+1)^2=12y(x+4)^2=3yрешите систему уравнений ​

Answer 1

a² - b² = (a - b)(a + b)

(4х+1)^2=12y

(x+4)^2=3y |*4   4(x + 4)^2 = 12y

вычитаем второе из первого

(4x + 1)^2 - (2(x + 4))^2 = 12y - 12y

(4x + 1 - 2x - 8)(4x + 1 + 2x + 8) = 0

(2x - 7)(6x + 9) = 0

x = 7/2

(4*7/2 + 1)^2 = (14 + 1)^2 = 15^2 = 225 = 12y

y = 225/12

x = -9/6 = -3/2

(4*(-3/2) + 1)^2 = (-6 + 1)^2 = 25 = 12y

y = 25/12

(-3/2, 25/12) (7/2, 225/12)

Answer 2

Let's expand the first equation:

(4x + 1)^2 = 16x^2 + 8x + 1 = 12y

And simplify the second equation by factoring out y:

(x + 4)^2 = (x + 4)(x + 4) = (x^2 + 8x + 16)y/3

Now we can substitute 12y from the first equation into the second equation:

(x + 4)^2 = (x^2 + 8x + 16)y/4

Multiplying both sides by 4 gives:

4(x + 4)^2 = (x^2 + 8x + 16)y

Substituting this into the first equation gives:

16x^2 + 8x + 1 = 4(x + 4)^2/3

Multiplying both sides by 3 gives:

48x^2 + 24x + 3 = 4(x + 4)^2

Dividing both sides by 4 gives:

12x^2 + 6x + 3/4 = (x + 4)^2

Taking the square root of both sides gives:

sqrt(12x^2 + 6x + 3/4) = x + 4

Subtracting x from both sides gives:

sqrt(12x^2 + 6x + 3/4) - x = 4

Squaring both sides gives:

12x^2 + 6x + 3/4 - 2x(sqrt(12x^2 + 6x + 3/4)) + x^2 = 16

Simplifying gives:

13x^2 - 2x(sqrt(12x^2 + 6x + 3/4)) + 13/4 = 0

This is a quadratic equation in terms of sqrt(12x^2 + 6x + 3/4). We can solve for this using the quadratic formula:

sqrt(12x^2 + 6x + 3/4) = (2x ± sqrt(4x^2 - 13/4))/13

Squaring both sides gives:

12x^2 + 6x + 3/4 = (4x^2 - 13/4 ± 4x(sqrt(4x^2 - 13/4)))/169

Multiplying both sides by 169 gives:

2028x^2 + 1014x + 51 = 16x^4 - 52x^2 + 13

Rearranging terms gives:

16x^4 - 2076x^2 + 1014x - 38 = 0

This is another quadratic equation in terms of x^2. We can solve for x^2 using the quadratic formula:

x^2 = (2076 ± sqrt(2076^2 - 4(16)(1014 - 38)))/(2(16))

x^2 = (2076 ± sqrt(4298496))/32

x^2 = (2076 ± 2074)/32

x^2 = 104/16 or x^2 = 4/16

x = ± sqrt(26)/2 or x = ± 1/2

Substituting these values of x into the second equation gives:

For x = sqrt(26)/2:

(x + 4)^2 = (26/2 + 8sqrt(26)/2 + 16)y/3

(9 + 4sqrt(26))y = (x + 4)^2 = (sqrt(26)/