Factorise 20x 2 9x 1

Factorising Quadratic Expressions: A Deep Dive into 20x² + 9x + 1

This article provides a comprehensive guide on how to factorise the quadratic expression 20x² + 9x + 1. We'll explore various methods, delve into the underlying mathematical principles, and address common difficulties encountered when factoring quadratic equations. Understanding this process is crucial for solving quadratic equations, simplifying algebraic expressions, and mastering more advanced mathematical concepts. This guide will equip you with the knowledge and skills to confidently tackle similar problems.

Understanding Quadratic Expressions

Before we begin factorising 20x² + 9x + 1, let's establish a foundational understanding of quadratic expressions. 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 is not equal to zero. In our example, a = 20, b = 9, and c = 1.

Factorising a quadratic expression means rewriting it as a product of two linear expressions. This process is the reverse of expanding brackets using the distributive property (often referred to as FOIL – First, Outer, Inner, Last). Finding the correct factors is key, and several methods can be used.

Method 1: The AC Method (for Factoring Trinomials)

The AC method is a systematic approach to factoring quadratic trinomials (expressions with three terms). It's particularly useful when the coefficient of x² (a) is not 1. Here's how it works for 20x² + 9x + 1:

1. Find the product AC: Multiply the coefficient of the x² term (a) by the constant term (c). In our case, AC = 20 * 1 = 20.

2. Find two numbers that add up to B and multiply to AC: We need two numbers that add up to the coefficient of the x term (b = 9) and multiply to 20. These numbers are 4 and 5 (4 + 5 = 9 and 4 * 5 = 20).

3. Rewrite the middle term: Rewrite the middle term (9x) as the sum of the two numbers found in step 2, using x as the variable. This gives us 20x² + 4x + 5x + 1.

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

   • (20x² + 4x) + (5x + 1)
   • 4x(5x + 1) + 1(5x + 1)

5. Factor out the common binomial: Notice that both terms now share the common binomial (5x + 1). Factor this out:

   • (5x + 1)(4x + 1)

Therefore, the factorised form of 20x² + 9x + 1 is (5x + 1)(4x + 1).

Method 2: Trial and Error

This method involves systematically trying different combinations of factors until you find the correct pair. While less systematic than the AC method, it can be quicker for simpler quadratics. For 20x² + 9x + 1:

1. Consider factors of the first term (20x²): The possible pairs are (1x, 20x), (2x, 10x), (4x, 5x), and their negatives.

2. Consider factors of the last term (1): The only pairs are (1, 1) and (-1, -1).

3. Test combinations: We need to find a combination that, when expanded using FOIL, results in the original expression. Let's try (4x + 1)(5x + 1):

   • (4x + 1)(5x + 1) = 20x² + 4x + 5x + 1 = 20x² + 9x + 1

This combination works, confirming that the factorised form is (4x + 1)(5x + 1) (Note: The order of the factors doesn't matter; (5x+1)(4x+1) is also correct).

Method 3: Using the Quadratic Formula (for Finding Roots)

While not directly factoring, the quadratic formula can help find the roots (solutions) of the equation 20x² + 9x + 1 = 0. These roots can then be used to construct the factored form. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For our equation:

• a = 20
• b = 9
• c = 1

Substituting these values into the formula yields:

x = [-9 ± √(9² - 4 * 20 * 1)] / (2 * 20) x = [-9 ± √(81 - 80)] / 40 x = [-9 ± √1] / 40 x = (-9 ± 1) / 40

This gives us two solutions:

• x = (-9 + 1) / 40 = -8/40 = -1/5
• x = (-9 - 1) / 40 = -10/40 = -1/4

These solutions correspond to the factors (5x + 1) and (4x + 1), respectively. If x = -1/5, then 5x + 1 = 0, and if x = -1/4, then 4x + 1 = 0. Therefore, the factored form is (5x + 1)(4x + 1).

Solving Quadratic Equations using Factorised Form

Once we have factorised the quadratic expression, we can use it to solve the corresponding quadratic equation (20x² + 9x + 1 = 0). Since the product of the two factors equals zero, either one of the factors must equal zero:

• 5x + 1 = 0 or 4x + 1 = 0

Solving for x in each equation gives us the solutions:

• 5x = -1 => x = -1/5
• 4x = -1 => x = -1/4

These are the roots of the quadratic equation.

The Significance of Factoring

Factoring quadratic expressions is a fundamental skill in algebra with broad applications. It's essential for:

• Solving quadratic equations: As demonstrated above, factoring allows us to find the solutions (roots) of a quadratic equation easily.
• Simplifying algebraic expressions: Factoring can significantly simplify more complex expressions, making them easier to manipulate and analyze.
• Sketching graphs of quadratic functions: The factored form reveals the x-intercepts (where the graph crosses the x-axis), which are crucial for accurately sketching the parabola.
• Further mathematical studies: A solid understanding of factoring is a prerequisite for more advanced mathematical topics like calculus and differential equations.

Frequently Asked Questions (FAQ)

Q: What if I can't find the factors easily?

A: If the trial and error method proves challenging, the AC method provides a more structured approach, guaranteeing you'll find the factors if they exist. Alternatively, the quadratic formula will always provide the roots, from which the factors can be derived.

Q: Are there other methods for factoring quadratics?

A: Yes, other less common methods exist, including completing the square and using specialized software or online calculators. However, the AC method and trial and error are generally sufficient for most problems.

Q: What if the quadratic expression cannot be factored?

A: Some quadratic expressions cannot be factored using integers. In these cases, the quadratic formula is still applicable to find the roots, and the expression might be expressed in a different form, potentially involving irrational numbers.

Q: What if the quadratic equation has only one solution (a repeated root)?

A: In such cases, the quadratic expression will be a perfect square trinomial, meaning it can be factored into the square of a binomial. For example, x² + 2x + 1 factors to (x + 1)².

Conclusion

Factorising quadratic expressions like 20x² + 9x + 1 is a core algebraic skill with wide-ranging applications. This article has demonstrated three methods – the AC method, trial and error, and utilizing the quadratic formula – to effectively factorise such expressions. Mastering these techniques will significantly enhance your ability to solve quadratic equations, simplify algebraic expressions, and progress in your mathematical studies. Remember to practice regularly and explore different approaches to find the method that best suits your learning style. The more you practice, the more intuitive and efficient you'll become at factoring quadratic expressions.