Certainly! Let’s factorize the given quadratic expression step by step.
Given expression:
[tex]\[ 4x^2 - 8x + 16 \][/tex]
Step 1: Factor out the greatest common factor (GCF) first, if there is any. In this case, the GCF of the coefficients [tex]\(4\)[/tex], [tex]\(-8\)[/tex], and [tex]\(16\)[/tex] is [tex]\(4\)[/tex].
[tex]\[ 4x^2 - 8x + 16 = 4(x^2 - 2x + 4) \][/tex]
Step 2: Now, we still need to attempt to factorize the quadratic expression inside the parentheses.
[tex]\[ x^2 - 2x + 4 \][/tex]
To do this, we usually look for two numbers that multiply to the constant term (+4) and add up to the coefficient of the linear term (-2).
However, since [tex]\(x^2 - 2x + 4\)[/tex] does not factorize nicely into integers, we verify that it indeed does not have real roots by using the discriminant method:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 4\)[/tex]:
[tex]\[ \Delta = (-2)^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 \][/tex]
Since the discriminant [tex]\( \Delta \)[/tex] is negative, the quadratic expression [tex]\(x^2 - 2x + 4\)[/tex] has no real roots and does not factorize further over the real numbers.
Therefore, the fully factorized form of the expression is:
[tex]\[ 4(x^2 - 2x + 4) \][/tex]
Thus, the factorized form of [tex]\(4x^2 - 8x + 16\)[/tex] is:
[tex]\[ 4(x^2 - 2x + 4) \][/tex]
