- The quadratic expression \(x^2 - 12x + 32\) is factored into the form \((x - p)(x - q)\).
- We identify that the coefficients require finding two numbers whose product is 32 and whose sum is 12.
- The numbers that satisfy these conditions are 4 and 8.
- The factored form is \((x - 4)(x - 8)\).
- Final Answer: \(\boxed{(x - 4)(x - 8)}\)
Explanation
- Problem Analysis and Setup
We are given the quadratic expression \(x^2 - 12x + 32\) and we need to factor it into the form \((x - p)(x - q)\). The expansion of this factored form is \(x^2 - (p + q)x + pq\). By comparing the coefficients of the given expression \(x^2 - 12x + 32\) to the expanded form, we must find two numbers \(p\) and \(q\) such that their sum is 12 and their product is 32.
- Finding the Constants p and q
We need to find two numbers that satisfy the following system of equations:
- \(p + q = 12\)
- \(pq = 32\)
To solve this, we can list the factors of 32 and check their sums:
- Factors of 32 are: (1, 32), (2, 16), and (4, 8).
- Checking the sums of these pairs:
- \(1 + 32 = 33\)
- \(2 + 16 = 18\)
- \(4 + 8 = 12\)
The pair (4, 8) matches our requirement that the sum must be 12. Therefore, \(p = 4\) and \(q = 8\).
- Conclusion and Verification
Substituting the identified values of \(p = 4\) and \(q = 8\) back into the factored form \((x - p)(x - q)\), we obtain \((x - 4)(x - 8)\). We can verify this result by expanding it:
\((x - 4)(x - 8) = x \cdot x - 8 \cdot x - 4 \cdot x + (-4) \cdot (-8) = x^2 - 8x - 4x + 32 = x^2 - 12x + 32\).
The result is correct. The blanks in the expression \((x - \_\_\_)(x - \_\_\_)\) are filled by 4 and 8.
Examples
Factoring quadratics is a foundational skill for solving equations in many real-world scenarios. For example, if an engineer is designing a parabolic arch for a bridge and needs to determine where the arch intersects the ground level, they set the quadratic model of the arch equal to zero. Factoring the expression makes it easy to find those specific points of intersection, allowing them to precisely determine the span and height required for a stable construction.