Fill in the blank below so that the resulting statement is true.
x squared minus 3 x minus 28equals​(xminus7​)(x___ ​)
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Part 1
x squared minus 3 x minus 28equalsleft parenthesis x minus 7 right parenthesis left parenthesis x
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To solve this problem, we need to factor the quadratic expression \( x^2 - 3x - 28 \) and match it to the given form \( (x - 7)(x \, \_ \, ) \).

We know one of the factors is \( (x - 7) \), and we need to find the missing factor that will complete the factorization.

### Step 1: Expand \( (x - 7)(x + \_ ) \)

Let the missing factor be \( (x + c) \), where \( c \) is a constant. Expanding this expression:

\[
(x - 7)(x + c) = x^2 + cx - 7x - 7c
\]
\[
= x^2 + (c - 7)x - 7c
\]

### Step 2: Compare with \( x^2 - 3x - 28 \)

We want \( x^2 + (c - 7)x - 7c \) to match \( x^2 - 3x - 28 \).

- For the linear term: \( c - 7 = -3 \), so \( c = 4 \).
- For the constant term: \( -7c = -28 \), which is true when \( c = 4 \).

Thus, the missing factor is \( (x + 4) \).

### Final Answer:
The complete factorization is \( (x - 7)(x + 4) \).

Would you like more details or have any questions?

Here are five related questions:
1. How do you check if a quadratic expression is factorable?
2. What is the relationship between the coefficients of a quadratic and its factored form?
3. How can you factor quadratic expressions when they are not easily factorable by inspection?
4. How does completing the square differ from factoring?
5. Can all quadratic equations be factored?

**Tip**: When factoring, always check by expanding the factors to ensure they match the original quadratic expression.

Fill in the blank below so that the resulting statement is true. x squared minus 3x minus 28 equals (x minus 7)(x ___ )

Learn how to factor the quadratic equation x^2 - 3x - 28 using algebraic methods. This step-by-step guide covers essential concepts like factoring and the Zero Product Property. Ideal for students in Grades 8-10.

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