Factorise 3x 2 5x 12

Factorising Quadratic Expressions: A Deep Dive into 3x² + 5x - 12

Factorising quadratic expressions is a fundamental skill in algebra. It's the process of rewriting a quadratic expression (like 3x² + 5x - 12) as a product of two simpler expressions. This seemingly simple task unlocks a world of possibilities in solving equations, simplifying expressions, and understanding the underlying structure of mathematical relationships. This article will guide you through the process of factorising 3x² + 5x - 12, exploring different methods and providing a deeper understanding of the underlying principles.

Understanding Quadratic Expressions

Before diving into the factorisation of 3x² + 5x - 12, let's refresh our understanding of quadratic expressions. A quadratic expression is an algebraic expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The term 'x²' represents the squared variable, 'x' is the linear term, and 'c' is the constant term. In our example, 3x² + 5x - 12, a = 3, b = 5, and c = -12.

Method 1: AC Method (Product-Sum Method)

This is a widely used method for factorising quadratic expressions, especially when the coefficient of x² (a) is not 1. The AC method relies on finding two numbers that satisfy specific product and sum conditions.

Steps:

1. Find the product 'ac': In our case, a = 3 and c = -12, so ac = 3 * (-12) = -36.

2. Find two numbers that multiply to 'ac' and add up to 'b': We need two numbers that multiply to -36 and add up to 5 (the coefficient of x). After some trial and error (or using a systematic approach), we find that the numbers 9 and -4 satisfy these conditions: 9 * (-4) = -36 and 9 + (-4) = 5.

3. Rewrite the middle term: Replace the middle term (5x) with the two numbers we found, expressing them as coefficients of x: 3x² + 9x - 4x - 12.

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

   3x(x + 3) - 4(x + 3)

5. Factor out the common binomial: Notice that both terms now share the common binomial factor (x + 3). Factor this out:

   (x + 3)(3x - 4)

Therefore, the factorised form of 3x² + 5x - 12 is (x + 3)(3x - 4).

Method 2: Trial and Error

This method involves systematically trying different combinations of binomial factors until you find the correct one. It’s more intuitive but can be time-consuming for more complex quadratics.

Steps:

1. Consider the factors of the leading coefficient (a) and the constant term (c): The factors of 3 are 1 and 3. The factors of -12 are numerous: (1, -12), (2, -6), (3, -4), (4, -3), (6, -2), (12, -1), and their negatives.

2. Test different combinations: We need to find a combination that results in the correct middle term (5x) when expanded. Let’s try some combinations:

   • (x + 1)(3x - 12): Expanding this gives 3x² - 9x - 12 – incorrect middle term.
   • (x + 2)(3x - 6): Expanding this gives 3x² + 0x -12 – incorrect middle term.
   • (x + 3)(3x - 4): Expanding this gives 3x² - 4x + 9x - 12 = 3x² + 5x - 12 – Correct!

Therefore, using trial and error, we again arrive at the factorised form (x + 3)(3x - 4).

Method 3: Using the Quadratic Formula (for finding roots, then factoring)

While not a direct factorisation method, the quadratic formula can help you find the roots of the quadratic equation 3x² + 5x - 12 = 0. Knowing the roots allows you to write the factorised form.

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

Substituting the values from our quadratic:

x = [-5 ± √(5² - 4 * 3 * -12)] / (2 * 3) x = [-5 ± √(25 + 144)] / 6 x = [-5 ± √169] / 6 x = [-5 ± 13] / 6

This gives two solutions:

x₁ = (-5 + 13) / 6 = 8/6 = 4/3 x₂ = (-5 - 13) / 6 = -18/6 = -3

The roots are 4/3 and -3. Knowing the roots, we can write the factorised form as:

a(x - x₁)(x - x₂) where 'a' is the coefficient of x².

So, 3(x - 4/3)(x + 3) = 3(x - 4/3)(x + 3) = (3x - 4)(x + 3)

This again yields the factorised form: (x + 3)(3x - 4).

A Deeper Look at the Mathematics

The success of these methods hinges on the fundamental theorem of algebra, which states that a polynomial of degree n has exactly n roots (solutions), possibly including complex numbers and repeated roots. Since our quadratic expression is of degree 2, it will have two roots. The factorisation process essentially reverses the expansion of two binomial expressions, allowing us to break down the quadratic into its constituent linear factors.

Why Factorisation is Important

Factorisation is a cornerstone of algebra for several reasons:

• Solving Quadratic Equations: Setting the factorised quadratic expression equal to zero allows us to easily solve for the values of 'x' (the roots) by setting each factor equal to zero.

• Simplifying Expressions: Factorisation simplifies complex algebraic expressions, making them easier to manipulate and understand.

• Finding the x-intercepts of Parabolas: In graphing quadratic functions, the roots (obtained through factorisation) represent the x-intercepts of the parabola.

• Foundation for Advanced Mathematics: Factorisation provides a solid foundation for more advanced mathematical concepts, including calculus, differential equations, and linear algebra.

Frequently Asked Questions (FAQ)

Q1: What if I can't find the numbers that satisfy the product-sum condition in the AC method?

A1: Double-check your calculations for 'ac' and 'b'. If you're still struggling, the quadratic expression might be prime (cannot be factorised using real numbers), or you might need to explore the use of the quadratic formula to find the roots and then construct the factored form.

Q2: Can I use any method to factorise a quadratic?

A2: Yes, you can try any of the methods described. The best method depends on your personal preference and the specific quadratic expression. The AC method is generally preferred for quadratics where 'a' is not 1, while trial and error can be quicker for simpler quadratics. The quadratic formula is a valuable tool when other methods fail or are overly time-consuming.

Q3: What happens if the discriminant (b² - 4ac) is negative?

A3: If the discriminant is negative, the quadratic equation has no real roots. The quadratic expression cannot be factorised using real numbers; instead, complex numbers will be involved in the factorisation.

Q4: Is there a shortcut for factorising specific types of quadratic expressions?

A4: Yes, there are shortcuts. For example, perfect square trinomials (like x² + 6x + 9 = (x+3)²) and difference of squares (like x² - 9 = (x-3)(x+3)) have specific patterns that allow for quicker factorisation.

Conclusion

Factorising quadratic expressions like 3x² + 5x - 12 is a crucial algebraic skill. Mastering various techniques like the AC method, trial and error, and using the quadratic formula provides you with a versatile toolkit to tackle a wide range of problems. Remember that practice is key to developing fluency and confidence in your factorisation skills. With consistent effort, you'll transform this seemingly challenging task into a straightforward and efficient process, opening doors to a deeper understanding of algebra and its numerous applications. The factorised form of 3x² + 5x - 12 is definitively (x + 3)(3x - 4), a result obtained through multiple approaches, reinforcing the understanding of the underlying principles.