To find the factored form of the quadratic expression [tex]\(x^2 + 12x - 64\)[/tex], we follow these steps:
1. Write the quadratic equation in standard form:
The given quadratic expression is already in the standard form [tex]\(ax^2 + bx + c\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(c = -64\)[/tex].
2. Identify two numbers that multiply to [tex]\(a \cdot c = 1 \cdot -64 = -64\)[/tex] and add to [tex]\(b = 12\)[/tex]:
We need two numbers whose product is [tex]\(-64\)[/tex] and whose sum is [tex]\(12\)[/tex].
3. Determine the pair of numbers:
By examining the factor pairs of [tex]\(-64\)[/tex], we find:
- [tex]\(1 \cdot -64 = -64\)[/tex] and [tex]\(1 + (-64) = -63\)[/tex]
- [tex]\(2 \cdot -32 = -64\)[/tex] and [tex]\(2 + (-32) = -30\)[/tex]
- [tex]\(4 \cdot -16 = -64\)[/tex] and [tex]\(4 + (-16) = -12\)[/tex]
- [tex]\(8 \cdot -8 = -64\)[/tex] and [tex]\(8 + (-8) = 0\)[/tex]
- [tex]\(-4 \cdot 16 = -64\)[/tex] and [tex]\(-4 + 16 = 12\)[/tex]
The pair [tex]\(-4\)[/tex] and [tex]\(16\)[/tex] satisfies both conditions.
4. Rewrite the middle term of the quadratic expression using the two identified numbers:
[tex]\[
x^2 + 12x - 64 = x^2 - 4x + 16x - 64
\][/tex]
5. Factor by grouping:
Group the terms to factor by grouping:
[tex]\[
(x^2 - 4x) + (16x - 64)
\][/tex]
6. Factor out the common factors in each group:
[tex]\[
x(x - 4) + 16(x - 4)
\][/tex]
7. Factor out the common binomial factor:
Notice that [tex]\((x - 4)\)[/tex] is a common factor:
[tex]\[
(x + 16)(x - 4)
\][/tex]
So, the factored form of the quadratic expression [tex]\(x^2 + 12x - 64\)[/tex] is [tex]\((x - 4)(x + 16)\)[/tex].
Thus, the correct answer is:
A. [tex]\((x-4)(x+16)\)[/tex]
