Question

Can someone explain how to solve this better? I am confused on how to 'complete the square.'
Find the center, vertices, and foci of the ellipse given by
4x2 - 16x + 9y2 + 18y - 11 = 0

Answer 1

a complete square is simply a perfect square ....

(ax+b)^2 = a^2 x^2 + 2ab x + b^2

if we compare that to what you have:

a^2 x^2 + 2ab x + b^2
4 x^2 -16 x + c

we see that a^2 = 4; 2ab = -16; and c=b^2

we need to determine the value of c so that adding and subtracting it (c-c=0) to the setup will give us a 'complete square'.

do the same process for the y parts. the rest is basic algebra

Answer 2

\[4x^2-16x+9y^2+18y-11=0\]Start by grouping all your x terms and your y terms together, and send the constant to the other side of the equation. \[(4x^2-16x)+(9y^2+18y)=11\]Once grouped, pull out the common multiple between each group. In this case it is 4 and 9 \[4(x^2-4x)+9(y^2+2y)=11\]Now you're trying to find a \(c\) value in order to complete a quadratic with both functions x and y. Remember, your quadratic function takes the form \(y=ax^2+bx+\color{red}c\) , and to find each c value, use the equation \(c=\left(\frac{b}{2}\right)^2\)
When you complete the square on one side of your function,you have to add the same c-value to the constant on the other side of the equation, but you have to remember that the value you're adding is multiplied to the multiple you've pulled out previously. (look at the highlighted red portions)\[4(x^2-4x\color{red}{+4})+9(y^2+2y\color{red}{+1})=11\color{red}{+16+9} \] Complete your square by writing your function in the form \[a\left(x\pm\color{red}{\frac{b}{2}}\right)+b\left(x\pm \color{red}{\frac{b}{2}}\right)=c\]\[4(x-\color{red}{2})^2+9(y+\color{red}{1})^2=36\]

Answer 3

@Jhannybean So how do we find the red 4 and red 1 on the left side of the equation? Is it because 16 and 9 are squares?

Answer 4

Maybe this video will help
https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/solving-quadratic-equations-by-completing-the-square

it shows how to complete the square , and then solve the quadratic (which you don't do here)