Final answer:
To solve the given quadratic equation, rearrange the terms, complete the square for y, and then isolate and solve for x. The result is an expression for x in terms of y.
Step-by-step explanation:
To solve the equation 36x^2 + 4y^2 + 8y - 104 = 0 using the method of completing the square, we'll focus on rearranging the terms and completing the square for both x and y separately. Here's a step-by-step process:
- Start with the original equation: 36x^2 + 4y^2 + 8y - 104 = 0.
- Rearrange the terms to group x's and y's: 36x^2 + 4(y^2 + 2y) - 104 = 0.
- Complete the square for the y-term: Add and subtract (2/2)^2 = 1 inside the parenthesis to get 4(y^2 + 2y + 1 - 1).
- Simplify: 36x^2 + 4(y + 1)^2 - 4 - 104 = 0.
- Merge constant terms: 36x^2 + 4(y + 1)^2 - 108 = 0.
- Isolate the x-term: 36x^2 = 108 - 4(y + 1)^2.
- Divide by 36: x^2 = 3 - (1/9)(y + 1)^2.
- Finally, solve for x: x = ±√[3 - (1/9)(y + 1)^2].
Now, you have an expression for x in terms of y, which you can use to find specific solutions for x given values of y, or vice versa.