The equation of a circle is 4x2 + 4y 2− 24x − 32y - 96 = 0.  Write the equation in standard form.

The standard form of a circle's equation is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius.

Given the equation 4x² + 4y² - 24x - 32y - 96 = 0, we can rewrite it as:

4(x² - 6x) + 4(y² - 8y) = 96

Now, to complete the square, we take half of the coefficients of x and y, square them and add to both sides. Half of -6 is -3, and (-3)² = 9. Half of -8 is -4, and (-4)² = 16.

So, we rewrite the equation as:

4[(x² - 6x + 9) - 9] + 4[(y² - 8y + 16) - 16] = 96

This simplifies to:

4[(x - 3)² - 9] + 4[(y - 4)² - 16] = 96

Further simplifying gives:

4(x - 3)² - 36 + 4(y - 4)² - 64 = 96

Combine like terms:

4(x - 3)² + 4(y - 4)² = 196

Divide through by 4 to get the equation in standard form:

(x - 3)² + (y - 4)² = 49

So, the center of the circle is at (3,4) and the radius is √49 = 7.