Final answer:
The quadratic equation 3x^2 - 12x + 9 = 0 can be solved by applying the quadratic formula, yielding solutions x = 1 and x = 3. These solutions indicate that the equation is indeed a perfect square, but the formula provides a systematic approach to reach the answer.
Step-by-step explanation:
To solve the quadratic equation 3x^2 - 12x + 9 = 0, we first identify the coefficients a=3, b=-12, and c=9. This equation takes the form ax^2 + bx + c = 0, where x represents the variable. Although you can use the quadratic formula to find the roots, we can also check if the equation is a perfect square.
If we divide the entire equation by 3, we get x^2 - 4x + 3 = 0, which is not readily recognizable as a perfect square. So, we proceed to use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a). Substituting the respective values, we have:
x = (12 ± √((-12)^2 - 4(3)(9))) / (2*3)
x = (12 ± √(144 - 108)) / 6
x = (12 ± √(36)) / 6
x = (12 ± 6) / 6
We then find the two potential solutions:
x = (12 + 6) / 6 = 18 / 6 = 3
x = (12 - 6) / 6 = 6 / 6 = 1
So, the solutions are x = 1 and x = 3. It appears that we do have a perfect square after all, so x could have been found more directly by factoring rather than applying the quadratic formula.