https://file.aiquickdraw.com/mathbot/file/1726442877636akjkd9ge.jpg

The problem in the image asks to find which of the following expressions is a factor of \( 3x^2 + 20x - 63 \):

1. \( x - 9 \)
2. \( 3x - 7 \)

We need to factor the quadratic expression \( 3x^2 + 20x - 63 \) and check if either or both of the given expressions are factors.

### Step 1: Factoring the quadratic

The quadratic expression is \( 3x^2 + 20x - 63 \). We are looking for two numbers that multiply to \( 3 \times (-63) = -189 \) and add to 20.

The numbers that work are 27 and -7, since:
\[
27 \times (-7) = -189 \quad \text{and} \quad 27 + (-7) = 20
\]

### Step 2: Splitting the middle term

We split the middle term \( 20x \) into \( 27x - 7x \):
\[
3x^2 + 27x - 7x - 63
\]

### Step 3: Grouping terms

Now, group the terms:
\[
(3x^2 + 27x) - (7x + 63)
\]

Factor out the common terms:
\[
3x(x + 9) - 7(x + 9)
\]

### Step 4: Factoring the common binomial

Now, factor out \( (x + 9) \):
\[
(3x - 7)(x + 9)
\]

### Step 5: Checking the expressions

- \( x - 9 \) is not a factor (since \( x + 9 \) is a factor, not \( x - 9 \)).
- \( 3x - 7 \) **is** a factor.

### Conclusion

The correct answer is **B. II only**.

Would you like a detailed explanation or have any other questions? 

Here are 5 related questions:
1. How do you factor more complex quadratics like \( ax^2 + bx + c \)?
2. What is the difference between factoring and solving quadratics?
3. How do you handle quadratics where no real solutions exist?
4. What other methods can be used to factor quadratics besides splitting the middle term?
5. How can you verify if a factorization is correct?

**Tip:** Always check your factorization by multiplying the factors back together to ensure you get the original expression.

Which of the following expressions is(are) a factor of 3x^2 + 20x - 63? I. x - 9 II. 3x - 7

Learn how to factor the quadratic expression 3x^2 + 20x - 63 and determine whether x - 9 or 3x - 7 are factors. This detailed step-by-step guide covers factoring by grouping and how to identify valid factors. Ideal for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation