Q: Convert 4x^2 + 4y^2 -8x -16y -29 = 0 into standard form.
I first started by pairing the like terms together, getting:
4x^2 -8x + 4y^2 -16y = 29
The next step, I'm a little confused. I know I need to complete the square, but I'm not sure how to fully factor out the above equation before doing so. Can someone help me getting the factored form of the above equation?
Thanks!
4x^2 -8x + 4y^2 -16y = 29
4(x^2 - 2x + ...) + 4(y^2 - 4y + ... ) = 29
4(x^2 - 2x + 1) + 4(y^2 - 4y + 4) = 29 + 4 + 16
4(x-1)^2 + 4(y-2)^2 = 49
divide by 4
(x-1)^2 + (y-2)^2 = 49/4
looks like a circle with centre at (1,2) and radius 7/2
To factor out the equation 4x^2 - 8x + 4y^2 - 16y = 29, we can start by rearranging the terms as follows:
4x^2 - 8x + 4y^2 - 16y - 29 = 0
Now, let's consider the x-terms and y-terms separately. We can rewrite the equation as:
(4x^2 - 8x) + (4y^2 - 16y) = 29
Now, let's factor out the common coefficients for each term:
4(x^2 - 2x) + 4(y^2 - 4y) = 29
Notice that we can complete the square for each of the quadratic terms inside the parentheses. To do so, we need to find the value we need to add and subtract to make the square perfect. In general, the value is (b/2)^2, where b is the coefficient of the linear term.
For x^2 - 2x, the coefficient of the linear term is -2 (b = -2), so the value we need to add and subtract is (-2/2)^2 = (-1)^2 = 1. Adding and subtracting 1, we get:
x^2 - 2x + 1 - 1
Similarly, for y^2 - 4y, the coefficient of the linear term is -4 (b = -4), so the value we need to add and subtract is (-4/2)^2 = (-2)^2 = 4. Adding and subtracting 4, we get:
y^2 - 4y + 4 - 4
Now, let's rewrite the equation:
4(x^2 - 2x + 1 - 1) + 4(y^2 - 4y + 4 - 4) = 29
Now, we can regroup the terms and factor the perfect squares:
4[(x - 1)^2 - 1] + 4[(y - 2)^2 - 4] = 29
Simplifying further:
4(x - 1)^2 - 4 + 4(y - 2)^2 - 16 = 29
Now, let's combine like terms:
4(x - 1)^2 + 4(y - 2)^2 - 20 = 29
To convert the equation to standard form, we need to move the constant term to the right side:
4(x - 1)^2 + 4(y - 2)^2 = 29 + 20
4(x - 1)^2 + 4(y - 2)^2 = 49
Finally, divide each term by the common factor (4) to obtain the standard form equation:
(x - 1)^2 + (y - 2)^2 = (49 / 4)
Therefore, the equation 4x^2 + 4y^2 - 8x - 16y - 29 = 0 can be written in standard form as (x - 1)^2 + (y - 2)^2 = (49 / 4).