Final answer:
To complete the square, rearrange terms, simplify by dividing, form perfect square trinomials, and solve for the constants which yield the completed square form.
Step-by-step explanation:
To complete the square for the quadratic equation 49x² + 16y² - 392x + 160y + 400 = 0, we need to rearrange the equation and form perfect square trinomials for x and y separately.
First, let's arrange the terms containing x and y together and leave the constant term on the other side:
49x² - 392x + 16y² + 160y = -400
Next, divide the equation by the coefficient of x² and y² to simplify the calculation:
x² - 8x + y² + 10y = -10
Now, take half of the coefficients of x and y, square them, and add to both sides of the equation:
x² - 8x + (8/2)² + y² + 10y + (10/2)² = -10 + (8/2)² + (10/2)²
x² - 8x + 16 + y² + 10y + 25 = -10 + 16 + 25
(x - 4)² + (y + 5)² = 31
This is the main answer, a completed square form of the given quadratic equation.