Question

The equation 4x2 – 24x + 4y2 + 72y = 76 is equivalent to

(1) 4(x – 3)2 + 4(y + 9)2 = 76
(2) 4(x – 3)2 + 4(y + 9)2 = 121
(3) 4(x – 3)2 + 4(y + 9)2 = 166
(4) 4(x – 3)2 + 4(y + 9)2 = 436
I know the answer is choice 1 but how do you get to that answer?

Answer 1

Option 4 is correct.

The equation
`4x^2 -24x + 4y^2 + 72y = 76` is equivalent to
`4(x-3)^2 + 4(y+9)^2 =436`

Step-by-step explanation:'

Given equation:
`4x^2 -24x + 4y^2 + 72y = 76`

First group the terms with x and those with y;

`(4x^2-24x)+(4y^2+72y) = 76`

Next, we complete the squares.

We can do this by adding a third term such that the x terms and the y terms are perfect squares.

For this we must either add the same value on the other side of the equation or subtract the same value on the same side so that the equality is maintained.

⇒
`4(x^2-6x) +4(y+18y) = 76`

or

`4(x^2 -6x +3^2 -3^2) + 4(y^2 +18y +9^2 -9^2) = 76`

`4(x^2-6x + 3^2) - 36 + 4(y^2+18y +9^2) - 324 = 76`

`4(x-3)^2 + 4(y+9)^2 - 360 =76`

Add 360 on both sides we get;

`4(x-3)^2 + 4(y+9)^2 =360 +76`

Simplify:

`4(x-3)^2 + 4(y+9)^2 =436`

Therefore, the given equation is equivalent to
`4(x-3)^2 + 4(y+9)^2 =436`