If \(p(x) = 4x^2 - 12x + 9\), what are the roots of \(p(x) = 0\)?

To find the roots of the polynomial (p(x) = 4x^2 - 12x + 9), we need to solve the equation (p(x) = 0). This can be done by using the quadratic formula, completing the square, or factoring, if applicable.

First, notice that the polynomial can be factored. We can express it as follows:

[

p(x) = 4x^2 - 12x + 9 = (2x - 3)^2

]

To confirm, expanding ((2x - 3)^2) gives:

[

(2x - 3)(2x - 3) = 4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9

]

This shows that (p(x)) can indeed be factored into ((2x - 3)^2). To find the roots, we set the factor equal to zero:

[

(2x - 3)^2 = 0

]

Taking the square root of both sides results in:

[

2x - 3 = 0

\