Certainly! Let's go through the step-by-step process of factorizing the given quadratic expression:
[tex]\[
16a^2 - 40a + 25
\][/tex]
1. Identify the quadratic expression:
The given expression is a quadratic polynomial in the form [tex]\(Ax^2 + Bx + C\)[/tex] where [tex]\(A = 16\)[/tex], [tex]\(B = -40\)[/tex], and [tex]\(C = 25\)[/tex].
2. Check for a perfect square trinomial:
A quadratic expression [tex]\(Ax^2 + Bx + C\)[/tex] can sometimes be factored into [tex]\((Dx + E)^2\)[/tex] where:
[tex]\[
D^2 = A \quad \text{and} \quad E^2 = C \quad \text{and} \quad 2DE = B
\][/tex]
3. Examine the coefficients:
We have [tex]\(A = 16\)[/tex], which is [tex]\(4^2\)[/tex], and [tex]\(C = 25\)[/tex], which is [tex]\(5^2\)[/tex].
So, we assume:
[tex]\[
D = 4 \quad \text{and} \quad E = 5
\][/tex]
4. Verify the middle term:
The middle term should be [tex]\(2DE\)[/tex]:
[tex]\[
B = 2 \cdot 4 \cdot 5 = 40
\][/tex]
5. Consider the sign of the middle term:
Since [tex]\(B = -40\)[/tex], [tex]\(E\)[/tex] should be negative. Thus:
[tex]\[
E = -5
\][/tex]
6. Construct and verify the factorization:
Thus, the expression becomes:
[tex]\[
(4a - 5)^2
\][/tex]
Let's confirm the factorization by expanding it back:
[tex]\[
(4a - 5)^2 = (4a - 5)(4a - 5) = 16a^2 - 20a - 20a + 25 = 16a^2 - 40a + 25
\][/tex]
The expansion matches the original expression.
Therefore, the factorized form of the quadratic expression [tex]\(16a^2 - 40a + 25\)[/tex] is:
[tex]\[
(4a - 5)^2
\][/tex]
