Question

Write 9x² - 36x = 4y² + 24y + 36 in standard form

this is a graph of a hyperbola, if this helps.
please include the steps on how you did it
Thank you!

Answer 1

9·x² - 36·x = 4·y² + 24·y + 36 in standard form is;

(x - 2)²/2² - (y + 3)²/3² = 1

Explanation:

The standard form of a hyperbola is given as follows;

(x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/b² - (x - h)²/a² = 1

The given equation is presented as follows;

9·x² - 36·x = 4·y² + 24·y + 36

By completing the square, we get;

(3·x - 6)·(3·x - 6) - 36 = (2·y + 6)·(2·y + 6)

(3·x - 6)² - 36 = (2·y + 6)²

(3·x - 6)² - (2·y + 6)² = 36

(3·x - 6)²/36 - (2·y + 6)²/36 = 36/36 = 1

(3·x - 6)²/6² - (2·y + 6)²/6² = 1

3²·(x - 2)²/6² - 2²·(y + 3)²/6² = 1

(x - 2)²/2² - (y + 3)²/3² = 1

The equation of the hyperbola is (x - 2)²/2² - (y + 3)²/3² = 1.