A rectangular enclosure has an area of 90 square yards. The length needs to be 6 yards longer than the width,x, to represent this she writes the equation x^2 + 6x =90 then she completes the square to help solve the equation for x

Question

Which equation is equivalent to her equation after she completes the square
(x+ 3)^2= 93
(x + 3)^2=99
(x + 6)^2=96
(x+5)^2=126

Answer 1

To complete the square for the equation \( x^2 + 6x = 90 \), we start by rearranging it:

1. Move the constant to the right side: \[ x^2 + 6x - 90 = 0 \] or, \[ x^2 + 6x = 90 \]

2. To complete the square, we take the coefficient of \( x \), which is 6, divide it by 2 to get 3, and then square it to get 9.

3. Add and subtract this square (9) inside the equation: \[ x^2 + 6x + 9 - 9 = 90 \] \[ (x + 3)^2 - 9 = 90 \]

4. Now add 9 to both sides: \[ (x + 3)^2 = 90 + 9 \] \[ (x + 3)^2 = 99 \]

Thus, the equivalent equation after completing the square is: \[ (x + 3)^2 = 99 \]

So, the correct answer is (x + 3)^2 = 99.