Math Problem Statement

4x^2-4x-48

Solution

To factor the quadratic expression 4x24x484x^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 4x24x^2, 4x-4x, and 48-48 is 4. 4(x2x12)4(x^2 - x - 12)

  2. Factor the quadratic expression inside the parentheses: We need to factor x2x12x^2 - x - 12.

    • Find two numbers that multiply to 12-12 (the constant term) and add to 1-1 (the coefficient of xx).
    • The numbers are 4-4 and 33 because 4×3=12-4 \times 3 = -12 and 4+3=1-4 + 3 = -1.
  3. Rewrite the quadratic expression: x24x+3x12x^2 - 4x + 3x - 12

  4. Group the terms: (x24x)+(3x12)(x^2 - 4x) + (3x - 12)

  5. Factor out the GCF from each group: x(x4)+3(x4)x(x - 4) + 3(x - 4)

  6. Factor out the common binomial factor: (x4)(x+3)(x - 4)(x + 3)

  7. Include the GCF that was factored out earlier: 4(x4)(x+3)4(x - 4)(x + 3)

So, the fully factored form of 4x24x484x^2 - 4x - 48 is: 4(x4)(x+3)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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Factoring

Formulas

Quadratic equation formula ax^2 + bx + c = 0

Theorems

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Suitable Grade Level

Grades 9-12