Answer :
To factor the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex], we can follow these steps:
1. Identify the Expression: We start with the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex].
2. Look for a Perfect Square Trinomial: A perfect square trinomial is one that can be written in the form [tex]\((x - b)^2 = x^2 - 2bx + b^2\)[/tex] or [tex]\((x + b)^2 = x^2 + 2bx + b^2\)[/tex]. In this case, our expression closely resembles a perfect square.
3. Check for the Form: Notice that:
- The first term [tex]\(a^2\)[/tex] is a perfect square, as it can be written as [tex]\((a)^2\)[/tex].
- The last term [tex]\(196\)[/tex] is a perfect square, since [tex]\(196 = 14^2\)[/tex].
4. Use the Perfect Square Trinomial Identity: The expression [tex]\(a^2 - 28a + 196\)[/tex] can be written as:
[tex]\[
(a - 14)^2 = a^2 - 2(14)a + 14^2
\][/tex]
5. Check:
- The middle term in the identity we've used is [tex]\(-2(14)a = -28a\)[/tex], which matches the middle term in our expression.
Therefore, the expression [tex]\(a^2 - 28a + 196\)[/tex] factors to [tex]\((a - 14)^2\)[/tex]. This means that the given expression is the square of the binomial [tex]\((a - 14)\)[/tex].
1. Identify the Expression: We start with the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex].
2. Look for a Perfect Square Trinomial: A perfect square trinomial is one that can be written in the form [tex]\((x - b)^2 = x^2 - 2bx + b^2\)[/tex] or [tex]\((x + b)^2 = x^2 + 2bx + b^2\)[/tex]. In this case, our expression closely resembles a perfect square.
3. Check for the Form: Notice that:
- The first term [tex]\(a^2\)[/tex] is a perfect square, as it can be written as [tex]\((a)^2\)[/tex].
- The last term [tex]\(196\)[/tex] is a perfect square, since [tex]\(196 = 14^2\)[/tex].
4. Use the Perfect Square Trinomial Identity: The expression [tex]\(a^2 - 28a + 196\)[/tex] can be written as:
[tex]\[
(a - 14)^2 = a^2 - 2(14)a + 14^2
\][/tex]
5. Check:
- The middle term in the identity we've used is [tex]\(-2(14)a = -28a\)[/tex], which matches the middle term in our expression.
Therefore, the expression [tex]\(a^2 - 28a + 196\)[/tex] factors to [tex]\((a - 14)^2\)[/tex]. This means that the given expression is the square of the binomial [tex]\((a - 14)\)[/tex].
