Answer :
To factor the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex], we can look for patterns or recognize it as a special type of trinomial.
1. Identify the expression: We start with the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex].
2. Look for a pattern: Notice that [tex]\(196\)[/tex] is a perfect square because [tex]\(14 \times 14 = 196\)[/tex]. This might suggest that we are dealing with a perfect square trinomial.
3. Consider perfect square trinomials: A perfect square trinomial is an expression of the form [tex]\((x - b)^2\)[/tex], which expands to [tex]\(x^2 - 2bx + b^2\)[/tex].
4. Compare with the general form:
- In the expression [tex]\(a^2 - 28a + 196\)[/tex], we have:
- [tex]\(x^2 = a^2\)[/tex],
- [tex]\(-2bx = -28a\)[/tex], and
- [tex]\(b^2 = 196\)[/tex].
5. Solve for [tex]\(b\)[/tex]:
- From [tex]\(-2b = -28\)[/tex], we solve for [tex]\(b\)[/tex] by dividing both sides by [tex]\(-2\)[/tex]:
[tex]\[
b = 14.
\][/tex]
6. Check the square of [tex]\(b\)[/tex]:
- Calculate [tex]\(b^2\)[/tex] to verify it matches:
[tex]\[
14^2 = 196,
\][/tex]
which is correct.
7. Write the factored form as a perfect square binomial:
- Therefore, the factorization of the expression is [tex]\((a - 14)^2\)[/tex].
So, the expression [tex]\(a^2 - 28a + 196\)[/tex] factors to [tex]\((a - 14)^2\)[/tex].
1. Identify the expression: We start with the quadratic expression [tex]\(a^2 - 28a + 196\)[/tex].
2. Look for a pattern: Notice that [tex]\(196\)[/tex] is a perfect square because [tex]\(14 \times 14 = 196\)[/tex]. This might suggest that we are dealing with a perfect square trinomial.
3. Consider perfect square trinomials: A perfect square trinomial is an expression of the form [tex]\((x - b)^2\)[/tex], which expands to [tex]\(x^2 - 2bx + b^2\)[/tex].
4. Compare with the general form:
- In the expression [tex]\(a^2 - 28a + 196\)[/tex], we have:
- [tex]\(x^2 = a^2\)[/tex],
- [tex]\(-2bx = -28a\)[/tex], and
- [tex]\(b^2 = 196\)[/tex].
5. Solve for [tex]\(b\)[/tex]:
- From [tex]\(-2b = -28\)[/tex], we solve for [tex]\(b\)[/tex] by dividing both sides by [tex]\(-2\)[/tex]:
[tex]\[
b = 14.
\][/tex]
6. Check the square of [tex]\(b\)[/tex]:
- Calculate [tex]\(b^2\)[/tex] to verify it matches:
[tex]\[
14^2 = 196,
\][/tex]
which is correct.
7. Write the factored form as a perfect square binomial:
- Therefore, the factorization of the expression is [tex]\((a - 14)^2\)[/tex].
So, the expression [tex]\(a^2 - 28a + 196\)[/tex] factors to [tex]\((a - 14)^2\)[/tex].
