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

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

• A. \(x = 0\)
• B. \(x = 3\)
• C. \(x = \frac{3}{2}\)
• D. \(x = -3\)

Answer: (C)

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 \

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

\