To factorize the given quadratic expression [tex]\( x^2 - 14x + 49 \)[/tex] and express it in the form [tex]\( (x + A)^B \)[/tex], we should follow these steps:
1. Identify the quadratic expression: The given expression is [tex]\( x^2 - 14x + 49 \)[/tex].
2. Recognize it as a perfect square trinomial:
First, we need to check if the expression can be rewritten as a perfect square trinomial. A perfect square trinomial takes the form [tex]\( (x + p)^2 \)[/tex], which expands to [tex]\( x^2 + 2px + p^2 \)[/tex].
3. Determine the constant term:
In our expression [tex]\( x^2 - 14x + 49 \)[/tex], the constant term is 49, which is the square of 7. To match our expression to a perfect square trinomial, we should check if the middle coefficient (-14) is twice the product of the square root of the constant term (7).
4. Verify the middle term:
The middle term in the trinomial should be [tex]\( -14x \)[/tex], which can be thought of as [tex]\( -2 \cdot 7 \cdot x \)[/tex] (i.e., [tex]\( 2px \)[/tex] where [tex]\( p = 7 \)[/tex]). This matches perfectly.
5. Write the trinomial as a perfect square:
Hence, the trinomial [tex]\( x^2 - 14x + 49 \)[/tex] can be rewritten as [tex]\( (x - 7)^2 \)[/tex].
6. Identify A and B:
Comparing the factored form [tex]\( (x - 7)^2 \)[/tex] with [tex]\( (x + A)^B \)[/tex], we can deduce the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- [tex]\( A = -7 \)[/tex], because [tex]\( x + A = x - 7 \)[/tex]
- [tex]\( B = 2 \)[/tex], because the exponent of the perfect square trinomial is 2.
So, the values are:
[tex]\[ A = -7 \][/tex]
[tex]\[ B = 2 \][/tex]
Therefore, the factorization of [tex]\( x^2 - 14x + 49 \)[/tex] is [tex]\( (x + A)^B \)[/tex] with [tex]\( A = -7 \)[/tex] and [tex]\( B = 2 \)[/tex].
