Factor Completely 3x 2 7x 6

Factoring Completely: A Deep Dive into 3x² + 7x - 6

Factoring quadratic expressions is a fundamental skill in algebra, crucial for solving equations, simplifying expressions, and understanding the behavior of functions. This article delves deep into the complete factorization of the quadratic expression 3x² + 7x - 6, exploring various methods and highlighting the importance of understanding the underlying principles. We'll move beyond simply finding the factors to understanding why certain methods work and how to apply them effectively in different scenarios.

Understanding Quadratic Expressions

Before diving into the factorization, let's solidify our understanding of quadratic expressions. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually 'x') is 2. It generally takes the form ax² + bx + c, where 'a', 'b', and 'c' are constants (numbers). In our case, a = 3, b = 7, and c = -6.

Key Terms:

• Coefficient: The numerical factor of a term (e.g., 3 in 3x², 7 in 7x).
• Constant: A term without a variable (e.g., -6).
• Factors: Numbers or expressions that multiply together to give a product. Factoring is the process of finding these factors.
• Roots (or Zeros): The values of x that make the quadratic expression equal to zero. These are closely related to the factors.

Method 1: Factoring by Trial and Error

This method involves systematically trying different combinations of factors until we find the pair that works. It relies on understanding the relationship between the coefficients and the constant term.

Steps:

1. Consider the factors of the leading coefficient (a): In our case, a = 3, and its factors are 1 and 3. These will be the coefficients of 'x' in our binomial factors.

2. Consider the factors of the constant term (c): c = -6. Its factors include (1, -6), (-1, 6), (2, -3), and (-2, 3).

3. Test different combinations: We need to find a combination of factors that, when multiplied out using the FOIL (First, Outer, Inner, Last) method, results in the original expression, 3x² + 7x - 6.

Let's try some combinations:

• (x + 1)(3x - 6): This expands to 3x² - 6x + 3x - 6 = 3x² - 3x - 6 (Incorrect)
• (x - 1)(3x + 6): This expands to 3x² + 6x - 3x - 6 = 3x² + 3x - 6 (Incorrect)
• (x + 2)(3x - 3): This expands to 3x² - 3x + 6x - 6 = 3x² + 3x - 6 (Incorrect)
• (x + 3)(3x - 2): This expands to 3x² - 2x + 9x - 6 = 3x² + 7x - 6 (Correct!)
• (x - 3)(3x + 2): This expands to 3x² + 2x - 9x -6 = 3x² - 7x -6 (Incorrect)

Therefore, the complete factorization of 3x² + 7x - 6 is (x + 3)(3x - 2).

Method 2: Factoring by the AC Method (Grouping)

This method is more systematic and less reliant on guesswork, especially when dealing with larger coefficients.

Steps:

1. Find the product of 'a' and 'c': ac = (3)(-6) = -18

2. Find two numbers that multiply to 'ac' and add to 'b': We need two numbers that multiply to -18 and add to 7. These numbers are 9 and -2.

3. Rewrite the middle term: Rewrite the expression 7x as 9x - 2x: 3x² + 9x - 2x - 6

4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   3x(x + 3) - 2(x + 3)

5. Factor out the common binomial factor: (x + 3)(3x - 2)

This confirms our result from the trial-and-error method.

Method 3: Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, even those that are difficult to factor. While it doesn't directly give the factored form, the roots can be used to determine the factors.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For our expression, a = 3, b = 7, and c = -6. Substituting these values into the formula gives:

x = [-7 ± √(7² - 4 * 3 * -6)] / (2 * 3) x = [-7 ± √(49 + 72)] / 6 x = [-7 ± √121] / 6 x = [-7 ± 11] / 6

This gives two solutions:

x₁ = (-7 + 11) / 6 = 4/6 = 2/3 x₂ = (-7 - 11) / 6 = -18/6 = -3

The roots are 2/3 and -3. These roots correspond to the factors (3x - 2) and (x + 3), respectively. Therefore, the factored form is (x + 3)(3x - 2).

Understanding the Relationship Between Roots and Factors

The connection between the roots of a quadratic equation and its factors is fundamental. If 'r₁' and 'r₂' are the roots of the quadratic equation ax² + bx + c = 0, then the factored form of the quadratic expression is a(x - r₁)(x - r₂).

In our case, the roots are 2/3 and -3. Therefore, the factored form is:

3(x - 2/3)(x + 3) = (3x - 2)(x + 3)

Applications of Factoring

Factoring quadratic expressions is a vital skill in numerous algebraic contexts:

• Solving Quadratic Equations: Setting the factored expression equal to zero allows you to find the roots (solutions) of the quadratic equation.

• Simplifying Rational Expressions: Factoring the numerator and denominator can lead to simplification by canceling common factors.

• Graphing Quadratic Functions: The factored form reveals the x-intercepts (where the graph crosses the x-axis) of the parabola.

• Calculus: Factoring is essential in techniques like finding derivatives and integrals.

Conclusion

Factoring the quadratic expression 3x² + 7x - 6 completely results in (x + 3)(3x - 2). We explored three different methods: trial and error, the AC method, and using the quadratic formula. Each method offers a different approach, and understanding the underlying principles connecting factors and roots strengthens your algebraic skills significantly. Mastering these techniques is crucial for success in algebra and its applications in higher-level mathematics and related fields. Remember to practice regularly to build fluency and confidence in factoring quadratic expressions.