completing the square?

Question

I need help completing the square of the following:

4x^2 + 9y^2 + 8x - 18y = 23

Thanks in advance!

Answer 1

First, handle the x and y parts separately:

4x²+8x + 9y²-18y = 23

Since the quadratic term of the first part is 4x², which is (2x)², we know that the completed square will be of the form (2x+k)², which is 4x² + 4kx + k². Comparing coefficients reveals 4k=8, so k=2, so to complete the square we need to add k², which is 4, so we have:

4x²+8x+4 + 9y²-18y = 27

Now factor:

(2x+2)² + 9y² - 18y = 27

For the second part, it will be of the form (3y+k)² = 9y² + 6ky + k². We have 6k=-18, so k=-3. Therefore, k²=9, so adding 9 to both sides:

(2x+2)² + 9y² - 18y + 9 = 36

Factoring:

(2x+2)² + (3y-3)² = 36

And we are done with the squares. However, we can proceed further to help interpret this figure:

2²(x+1)² + 3²(y-1)² = 6²

Dividing by 6²:

(x+1)²/3² + (y-1)²/2² = 1

This is the standard form for the equation of an ellipse centered at (-1, 1), having a horizontal semimajor axis of length 3 and semiminor axis of length 2.

Answer 2

4x^2 + 9y^2 + 8x - 18y = 23
(4x^2 +8x + 4 ) - 4 + ( 9y^2 -18y +9 ) -9 =23
4( x^2 +2x + 1 ) +9( y^2 -2y +1)=23+4+9
4(x +1)^2 +9( y-1)^2 = 36
2^2(x +1)^2+3^2( y-1)^2 = 6^2

Answer 3

4x^2 + 9y^2 + 8x - 18y = 23
4x^2 + 8x + 9y^2 - 18y = 23
4(x^2 + 2x) + 9(y^2 - 2y) = 23
4[(x+1)^2 - 1] + 9[(y-1)^2 - 1] = 23
4(x+1)^2 - 4 + 9(y-1)^2 - 9 = 23
4(x+1)^2 + 9(y-1)^2 = 23 + 4 + 9
4(x+1)^2 + 9(y-1)^2 = 36

Answer 4

(4x^2 + 8x) + (9y^2 - 18y) = 23
{(2x)^2 + 2(2x)(2) + 2^2} + {(3y)^2 - 2(3y)(3) + 3^2} = 23 + 4 + 9
(2x + 2)^2 + (3y - 3)^2 = 36
{2(x + 1)}^2 + {3(y - 1)}^2 = 6^2

This is the equation of an ellipse with centre at (-1,1) and major and minor axes parallel to x- and y- axes respectively. The length of major axis is 6 units and that of minor axis is 4 units.

Answer 5

4x^2+8x+4+9y^2-18y+9=36
(2x+2)^2 +(3y-3)^2=36

x=sqr [36-(3y-3)]-2 /2

Answer 6

you need middle terms to be xy not x and y, this has no solution.

Answer 7

There is no way to figure it out

Answer 8

this is undefine