To factor the quadratic expression \( 4x^2 - 4x - 48 \), let's start by factoring out the greatest common factor (GCF) first.

### Steps:

1. **Factor out the GCF:**
   The GCF of the terms \(4x^2\), \(-4x\), and \(-48\) is 4.
   \[
   4(x^2 - x - 12)
   \]

2. **Factor the quadratic expression inside the parentheses:**
   We need to factor \(x^2 - x - 12\).

   - Find two numbers that multiply to \(-12\) (the constant term) and add to \(-1\) (the coefficient of \(x\)).
   - The numbers are \(-4\) and \(3\) because \(-4 \times 3 = -12\) and \(-4 + 3 = -1\).

3. **Rewrite the quadratic expression:**
   \[
   x^2 - 4x + 3x - 12
   \]

4. **Group the terms:**
   \[
   (x^2 - 4x) + (3x - 12)
   \]

5. **Factor out the GCF from each group:**
   \[
   x(x - 4) + 3(x - 4)
   \]

6. **Factor out the common binomial factor:**
   \[
   (x - 4)(x + 3)
   \]

7. **Include the GCF that was factored out earlier:**
   \[
   4(x - 4)(x + 3)
   \]

So, the fully factored form of \( 4x^2 - 4x - 48 \) is:
\[
4(x - 4)(x + 3)
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:

1. How do you identify the greatest common factor in a polynomial?
2. Can you always factor out a GCF before factoring the rest of the expression?
3. How do you factor trinomials with different leading coefficients?
4. What if the quadratic expression doesn’t have integer roots?
5. How do you solve a factored quadratic equation?
6. What is the significance of the constant term in factoring?
7. How do you check your work after factoring a quadratic?
8. How do you factor a quadratic expression when it is not easily factorable?

**Tip:** Always start by factoring out the greatest common factor (GCF) before attempting to factor the quadratic expression itself. This simplifies the expression and makes the factorization process easier.

Learn how to factor the quadratic expression 4x^2 - 4x - 48 with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratic expressions and the quadratic equation formula. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation