Question

If using the method of completing the square to solve the quadratic equation x, squared, plus, 3, x, plus, 9, equals, 0x 2 3x 9=0, which number would have to be added to "complete the square"?

Answer 1

To complete the square for the equation x² + 3x + 9 = 0, add (3/2)², which is 9/4 or 2.25, to form the perfect square trinomial (x + 3/2)², resulting in the new equation (x + 3/2)² = -27/4.

Step-by-step explanation:

To complete the square for the quadratic equation x² + 3x + 9 = 0, we first need to find a number that when added to x² + 3x, turns it into a perfect square trinomial. The process involves taking half of the coefficient of x, which is 3/2, and then squaring it to get (3/2)², which is 2.25 or 9/4. This is the number that we need to add to both sides of the equation to complete the square.

To illustrate, we transform the equation into a square by adding and subtracting 9/4:

x² + 3x + (9/4) - (9/4) + 9 = 0

Simplifying, we combine like terms:

x² + 3x + (9/4) = -9 + (9/4)

And we can then write the left side as a squared term:

(x + 3/2)² = -27/4