Question

Select the correct answer. A quadratic equation is shown below: 9x^2 - 108x + 216 = 0. Which of the following is an equivalent form of the above equation?

A. (x - 6)^2 = 12
B. 9(x - 12)^2 = 216
C. 9(x - 6)^2 = 60
D. (x - 6)^2 = 300

Answer 1

The equivalent form of the quadratic equation 9x^2 - 108x + 216 = 0 after completing the square is 9(x - 6)^2 = 216, which corresponds to option B from the given choices.

Step-by-step explanation:

To determine the equivalent form of the quadratic equation 9x^2 - 108x + 216 = 0, we can try factoring or completing the square. Completing the square involves forming a perfect square trinomial on one side of the equation. Since the equation is already in the form ax^2 + bx + c = 0, we can start by factoring out the common factor of 9:

9(x^2 - 12x + 24) = 0

Now we need to turn the expression x^2 - 12x + 24 into a perfect square. The square of half the coefficient of x, which is 6, is 36. We notice that 24 is not equal to 36, so we look for an option where we add 36 inside the parenthesis and adjust the constant outside to maintain equality:

9(x^2 - 12x + 36) = 9(24) = 216

This simplifies to 9(x - 6)^2 = 216, which is equivalent to the original equation given. Among the options provided, option B is the equivalent form of the quadratic equation.