To determine which expression is equivalent to [tex]\(4x^2 - 36x + 81\)[/tex], we need to factor the quadratic polynomial.
We start with the quadratic expression:
[tex]\[ 4x^2 - 36x + 81 \][/tex]
To factor this, we look for a common factor or use methods such as factoring by grouping. In this case, we can recognize that the quadratic can be written in the form [tex]\((ax + b)^2\)[/tex].
The steps to factor the quadratic are as follows:
1. Write the quadratic in the standard form: [tex]\( ax^2 + bx + c \)[/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = -36 \)[/tex], and [tex]\( c = 81 \)[/tex].
2. Identify if it forms a perfect square trinomial:
A perfect square trinomial has the form [tex]\((mx + n)^2 = m^2x^2 + 2mnx + n^2\)[/tex].
3. Compare with the perfect square form:
We need [tex]\( m^2 = 4 \)[/tex], [tex]\( 2mn = -36 \)[/tex], and [tex]\( n^2 = 81 \)[/tex].
4. Solve for [tex]\( m \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ m^2 = 4 \Rightarrow m = 2 \text{ (since } m \text{ could also be } -2) \][/tex]
[tex]\[ n^2 = 81 \Rightarrow n = 9 \text{ (since } n \text{ could also be } -9) \][/tex]
5. Check the middle term [tex]\( 2mn = -36 \)[/tex]:
Substitute [tex]\( m = 2 \)[/tex] and [tex]\( n = -9 \)[/tex] into [tex]\( 2mn \)[/tex]:
[tex]\[ 2(2)(-9) = -36 \][/tex]
6. Rewrite the quadratic using [tex]\( m \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ 4x^2 - 36x + 81 = (2x - 9)^2 \][/tex]
Thus, the factored form of the quadratic expression [tex]\(4x^2 - 36x + 81\)[/tex] is:
[tex]\[ (2x - 9)^2 \][/tex]
Therefore, the corresponding answer from the given options is:
B. [tex]\((2x-9)(2x-9)\)[/tex]
