Factorise 6x 2 5x 6

Factorising the Quadratic Expression 6x² + 5x - 6: A complete walkthrough

Factorising quadratic expressions is a fundamental skill in algebra. Because of that, understanding this process is crucial for solving quadratic equations, simplifying algebraic expressions, and tackling more advanced mathematical concepts. This article provides a thorough explanation of how to factorise the quadratic expression 6x² + 5x - 6, covering various methods and delving into the underlying mathematical principles. We'll explore different approaches, ensuring you grasp the core concepts and can confidently factorise similar expressions.

Introduction: Understanding 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. Factorising a quadratic expression means rewriting it as a product of two linear expressions. In our case, we aim to factorise 6x² + 5x - 6 into two expressions that, when multiplied, give us the original quadratic.

Method 1: The AC Method (Product-Sum Method)

This method is a systematic approach to factorising quadratic expressions. It's particularly helpful when the coefficient of x² (a) is not 1.

Steps:

1. Find the product AC: In our expression, 6x² + 5x - 6, a = 6 and c = -6. That's why, AC = 6 * (-6) = -36.

2. Find two numbers that add up to B and multiply to AC: We need two numbers that add up to 5 (the coefficient of x, which is b) and multiply to -36. These numbers are 9 and -4 (9 + (-4) = 5 and 9 * (-4) = -36).

3. Rewrite the middle term: Replace the middle term (5x) with the two numbers we found: 9x and -4x. Our expression now becomes 6x² + 9x - 4x - 6.

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

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

5. Factor out the common binomial: Notice that (2x + 3) is common to both terms. Factor it out:

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

So, the factorised form of 6x² + 5x - 6 is (2x + 3)(3x - 2).

Method 2: Trial and Error

This method involves testing different combinations of factors until you find the correct pair. It relies on understanding how the multiplication of binomials works.

Steps:

1. Consider factors of the leading coefficient (a): The factors of 6 are 1, 2, 3, and 6.

2. Consider factors of the constant term (c): The factors of -6 are 1, -1, 2, -2, 3, -3, 6, -6.

3. Test combinations: We need to find a combination of factors that, when multiplied and combined, result in the middle term (5x). Let's try some combinations:

   • (x + 1)(6x - 6): This expands to 6x² - 6x + 6x -6 = 6x² - 6, which is incorrect.
   • (x + 2)(6x - 3): This expands to 6x² -3x + 12x - 6 = 6x² + 9x -6, which is also incorrect.
   • (2x + 3)(3x - 2): This expands to 6x² - 4x + 9x - 6 = 6x² + 5x - 6. This is the correct factorisation!

While this method can be quicker for some, it can be time-consuming and less systematic, especially for more complex quadratic expressions.

Method 3: Using the Quadratic Formula (Indirect Factorisation)

The quadratic formula provides the roots of a quadratic equation (ax² + bx + c = 0). These roots can then be used to find the factors.

Continue exploring with our guides on write your answer without parentheses and who were the members of the jacobin club.

The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

For our expression, a = 6, b = 5, and c = -6. Plugging these values into the formula:

x = [-5 ± √(5² - 4 * 6 * -6)] / (2 * 6) x = [-5 ± √(25 + 144)] / 12 x = [-5 ± √169] / 12 x = [-5 ± 13] / 12

This gives us two solutions:

x₁ = (-5 + 13) / 12 = 8/12 = 2/3 x₂ = (-5 - 13) / 12 = -18/12 = -3/2

The factors are then of the form (x - x₁) and (x - x₂):

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

To get rid of the fractions, multiply each factor by the denominator:

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

That's why, the factorised form is (3x - 2)(2x + 3) which is equivalent to the result obtained using the other methods.

Explanation of the Mathematical Principles

The success of all these methods hinges on the distributive property of multiplication (often referred to as the FOIL method – First, Outer, Inner, Last). When we multiply two binomials like (ax + b)(cx + d), we get:

acx² + adx + bcx + bd

This results in a quadratic expression. Now, factorising reverses this process, finding the original binomials that, when multiplied, produce the quadratic. The AC method systematically works backwards from the expanded form to find the appropriate binomials, while trial and error directly seeks the correct combination. The quadratic formula directly solves for the roots of the corresponding quadratic equation, allowing indirect determination of the factors.

Frequently Asked Questions (FAQ)

• Q: What if the quadratic expression cannot be factorised easily?

  A: Some quadratic expressions have roots that are irrational or complex numbers. In such cases, the quadratic formula is the most reliable method, even though the resulting factors might involve irrational or complex numbers.

• Q: Is there only one correct way to factorise a quadratic expression?

  A: Essentially, yes. g.Still, while the order of the factors can be reversed (e. , (2x + 3)(3x - 2) is the same as (3x - 2)(2x + 3)), there’s only one set of unique linear expressions that will multiply to give the original quadratic.

• Q: Why is factorising quadratic expressions important?

  A: Factorising is a crucial step in solving quadratic equations, simplifying more complex algebraic expressions, and understanding the behaviour of quadratic functions. It's also a foundation for more advanced topics in mathematics.

• Q: What if the coefficient of x² is 1?

  A: If a = 1, the process simplifies significantly. You only need to find two numbers that add up to b and multiply to c. Take this: factoring x² + 5x + 6 would involve finding two numbers that add to 5 and multiply to 6 (which are 2 and 3), resulting in (x + 2)(x + 3).

Conclusion: Mastering Quadratic Factorisation

Factorising quadratic expressions like 6x² + 5x - 6 is a fundamental skill in algebra. Consider this: with enough practice, you will develop an intuition for finding the right combinations, making the process faster and more efficient. This article explored three different methods – the AC method, trial and error, and using the quadratic formula – illustrating their application and highlighting the underlying mathematical principles. Because of that, by understanding these methods and practicing regularly, you can confidently factorise a wide range of quadratic expressions and solve related problems. Consider this: remember to choose the method that best suits your understanding and the complexity of the expression. Consistent practice will solidify your understanding and make factorisation a straightforward process. Remember, mastering this skill is a significant step towards success in higher-level mathematics.