Factorise 6x 2 5x 6

Factorising the Quadratic Expression 6x² + 5x - 6: A Comprehensive Guide

Factorising quadratic expressions is a fundamental skill in algebra. Understanding how to factorise allows you to simplify expressions, solve quadratic equations, and delve deeper into more complex mathematical concepts. This article will provide a thorough explanation of how to factorise the quadratic expression 6x² + 5x - 6, covering multiple methods and providing insights into the underlying mathematical principles. We'll explore both the trial-and-error method and the more systematic method using the AC method, ensuring you gain a complete understanding.

Understanding Quadratic Expressions

Before we delve into the factorisation process, let's clarify what a quadratic expression is. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. It generally takes the form ax² + bx + c, where a, b, and c are constants, and a ≠ 0. In our example, 6x² + 5x - 6, a = 6, b = 5, and c = -6. Our goal is to rewrite this expression as a product of two linear expressions (expressions of degree one).

Method 1: Trial and Error (Factorisation by Inspection)

This method involves systematically trying different combinations of factors of 'a' and 'c' until we find a pair that produces the middle term 'b' when expanded. It relies on intuition and familiarity with multiplication patterns.

1. Identify factors of 'a' and 'c':

   • The factors of 'a' (6) are (1, 6), (2, 3), (3,2), (6,1), (-1,-6), (-2,-3), (-3,-2), (-6,-1).
   • The factors of 'c' (-6) are (1, -6), (2, -3), (3, -2), (6, -1), (-1, 6), (-2, 3), (-3, 2), (-6, 1).

2. Test different combinations: We need to find a combination that, when multiplied and added, yields the middle term 'b' (5). Let's try some combinations:

   • (2x + 3)(3x - 2): Expanding this gives 6x² - 4x + 9x - 6 = 6x² + 5x - 6. This works!

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

Method 2: AC Method (Product-Sum Method)

The AC method provides a more systematic approach to factorising quadratic expressions, especially when dealing with larger numbers or when trial and error becomes cumbersome.

1. Find the product 'ac': In our case, a = 6 and c = -6, so ac = 6 * (-6) = -36.

2. Find two numbers that multiply to 'ac' and add to 'b': We need two numbers that multiply to -36 and add to 5. 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: 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 factor: Notice that (2x + 3) is a common factor in both terms. Factor it out:

   • (2x + 3)(3x - 2)

Therefore, using the AC method, we again arrive at the factorised form: (2x + 3)(3x - 2).

Mathematical Explanation: Why Factorisation Works

The fundamental principle behind factorising quadratic expressions lies in the distributive property of multiplication. The distributive property states that a(b + c) = ab + ac. When we factorise a quadratic expression, we are essentially reversing this process. We're finding two expressions that, when multiplied together using the distributive property, result in the original quadratic expression.

Solving Quadratic Equations using Factorisation

Once we have factorised a quadratic expression, we can use it to solve quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0. To solve it, we set the factorised expression equal to zero and solve for x:

(2x + 3)(3x - 2) = 0

This equation is true if either (2x + 3) = 0 or (3x - 2) = 0. Solving these linear equations gives us the solutions:

• 2x + 3 = 0 => 2x = -3 => x = -3/2
• 3x - 2 = 0 => 3x = 2 => x = 2/3

Therefore, the solutions to the quadratic equation 6x² + 5x - 6 = 0 are x = -3/2 and x = 2/3.

Frequently Asked Questions (FAQ)

Q: What if I can't find factors easily using the trial-and-error method?

A: If trial and error proves difficult, the AC method provides a more systematic and reliable approach. It’s particularly helpful when dealing with larger numbers or more complex quadratic expressions.

Q: Can all quadratic expressions be factorised?

A: No. Some quadratic expressions cannot be factorised using integers. In these cases, you might need to use the quadratic formula to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that provides the solutions to any quadratic equation of the form ax² + bx + c = 0. The formula is: x = [-b ± √(b² - 4ac)] / 2a.

Q: Is there a way to check if my factorisation is correct?

A: Yes! Always expand your factorised expression to ensure it matches the original quadratic expression. If they are identical, your factorisation is correct.

Conclusion

Factorising quadratic expressions is a crucial skill in algebra. This article has explored two methods: trial and error and the AC method. Both methods lead to the same result: (2x + 3)(3x - 2) for the expression 6x² + 5x - 6. Understanding these methods allows you to simplify expressions, solve quadratic equations, and build a stronger foundation in algebra. Remember to practice regularly to become proficient in factorisation and to confidently tackle more challenging algebraic problems in the future. The ability to factorise efficiently and accurately is essential for success in higher-level mathematics. Don't be afraid to explore both methods and choose the one that best suits your learning style and the complexity of the problem at hand. The more you practice, the more intuitive and efficient you will become.