Complete the factoring. x^2 – 12x + 32 = (x – 8)( )



To complete the factoring of the expression \( x^2 - 12x + 32 \), we need to find the second factor in the form \( (x - c) \).

The given factor is \( (x - 8) \). Let's proceed with the steps:

### Step 1: Identify the missing factor
We start with the factored form:

\[
x^2 - 12x + 32 = (x - 8)(x - c)
\]

### Step 2: Expand the expression
Expand the right-hand side to match the original quadratic expression:

\[
(x - 8)(x - c) = x^2 - cx - 8x + 8c = x^2 - (c + 8)x + 8c
\]

### Step 3: Match coefficients
Compare this with the original quadratic expression \( x^2 - 12x + 32 \). 

From the expansion:

- The coefficient of \( x \) gives us \( -(c + 8) = -12 \).
- The constant term gives us \( 8c = 32 \).

### Step 4: Solve for \( c \)
1. Solve \( 8c = 32 \) for \( c \):

\[
c = \frac{32}{8} = 4
\]

2. Verify by checking the linear term:

\[
c + 8 = 12 \quad \text{(which matches the coefficient of \( x \) in the original expression)}
\]

### Final Factored Form
Thus, the complete factorization of \( x^2 - 12x + 32 \) is:

\[
x^2 - 12x + 32 = (x - 8)(x - 4)
\]

Would you like any further details or have any questions?

Here are some related questions you might find helpful:
1. How do you factor quadratic expressions using the quadratic formula?
2. Can every quadratic expression be factored into two binomials?
3. What is the difference between factoring by grouping and simple factoring?
4. How can you recognize if a quadratic expression is a perfect square trinomial?
5. How do you factor more complex quadratic expressions with leading coefficients greater than 1?

**Tip:** After factoring a quadratic expression, always expand the factors to check that they give you the original expression. This ensures that the factorization is correct.

Learn how to complete the factoring of the quadratic expression x^2 – 12x + 32. This problem involves identifying the correct factors and verifying the solution. Suitable for high school students and those preparing for algebra exams.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation