Math Problem Statement

Factor the quadratic expression: 2x^2 - x - 6

Solution

To factor the quadratic expression 2x2x62x^2 - x - 6, we can follow these steps:

  1. Identify the coefficients: For the expression 2x2x62x^2 - x - 6, we have:

    • a=2a = 2
    • b=1b = -1
    • c=6c = -6
  2. Multiply aa and cc: Calculate 2×(6)=122 \times (-6) = -12.

  3. Find factors of -12 that add up to b=1b = -1:

    • The factors of -12 that add up to -1 are 4-4 and 33, because 4+3=1-4 + 3 = -1.
  4. Rewrite the middle term: Rewrite x-x as 4x+3x-4x + 3x, so the expression becomes: 2x24x+3x62x^2 - 4x + 3x - 6

  5. Factor by grouping:

    • Group terms: (2x24x)+(3x6)(2x^2 - 4x) + (3x - 6)
    • Factor out the common factors in each group:
      • 2x(x2)+3(x2)2x(x - 2) + 3(x - 2)
  6. Factor out the common binomial:

    • Combine the terms to get: (2x+3)(x2)(2x + 3)(x - 2)

So, the factored form of 2x2x62x^2 - x - 6 is: (2x+3)(x2)(2x + 3)(x - 2)

Would you like further explanation on any of the steps?


Here are five related questions to expand on this:

  1. How would you factor a quadratic expression with a positive constant term?
  2. Can you use the same method to factor a trinomial with a different leading coefficient?
  3. How does the factoring process change if the middle term is positive?
  4. What are alternative methods to factor a quadratic if it’s not easily factorable by grouping?
  5. How would you verify the factors are correct?

Tip: Always check your answer by expanding the factored form to ensure it matches the original expression.

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Math Problem Analysis

Mathematical Concepts

Algebra
Factoring
Quadratic Equations

Formulas

ax^2 + bx + c = 0
Factoring by grouping

Theorems

Factoring by grouping

Suitable Grade Level

Grades 8-10