How To Factorise A Quadratic

How to Factorise a Quadratic: A Comprehensive Guide

Factoring quadratics is a fundamental skill in algebra, crucial for solving quadratic equations, simplifying expressions, and understanding many mathematical concepts that build upon it. This comprehensive guide will walk you through the process of factorising quadratics, covering various methods and providing ample examples to solidify your understanding. Whether you're a beginner struggling with the basics or looking to refine your skills, this guide will equip you with the knowledge and confidence to tackle any quadratic factorisation problem. We'll explore different techniques, addressing common challenges and highlighting shortcuts along the way.

Understanding Quadratics

Before diving into the methods of factorisation, let's establish a clear understanding of what a quadratic expression is. 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), and a is not equal to zero (otherwise it wouldn't be a quadratic). The goal of factorising a quadratic is to rewrite it as a product of two linear expressions (expressions of degree one). This product will typically look like this:

(px + q)(rx + s)

where p, q, r, and s are constants. When you expand this expression using the FOIL method (First, Outer, Inner, Last), you should obtain the original quadratic expression.

Method 1: Factoring by Inspection (Simple Quadratics)

This method is best suited for simple quadratics where a = 1. This means the quadratic is of the form:

x² + bx + c

The strategy involves finding two numbers that add up to b (the coefficient of x) and multiply to c (the constant term).

Example: Factorise x² + 5x + 6

1. Find the factors: We need two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3 (2 + 3 = 5 and 2 * 3 = 6).

2. Write the factorised form: The factorised form is then (x + 2)(x + 3).

Example: Factorise x² - 7x + 12

1. Find the factors: We need two numbers that add up to -7 and multiply to 12. These numbers are -3 and -4 (-3 + -4 = -7 and -3 * -4 = 12).

2. Write the factorised form: The factorised form is (x - 3)(x - 4).

Important Note: If the constant term (c) is positive, both numbers will have the same sign (both positive or both negative), matching the sign of the coefficient of x (b). If c is negative, the two numbers will have opposite signs.

Method 2: Factoring by Grouping (General Quadratics)

This method is applicable to all quadratics, including those where a ≠ 1. It involves breaking down the middle term (bx) into two terms whose coefficients add up to b and whose product is equal to a * c*.

Example: Factorise 2x² + 7x + 3

1. Find the product ac: a * c = 2 * 3 = 6

2. Find the factors of ac: We need two numbers that add up to 7 (the coefficient of x) and multiply to 6. These numbers are 6 and 1 (6 + 1 = 7 and 6 * 1 = 6).

3. Rewrite the middle term: Rewrite 7x as 6x + 1x (or 1x + 6x). The quadratic becomes 2x² + 6x + 1x + 3.

4. Factor by grouping: Group the terms in pairs and factor out the common factor from each pair: 2x(x + 3) + 1(x + 3)

5. Factor out the common binomial: Notice that (x + 3) is a common factor. Factor it out: (x + 3)(2x + 1)

Therefore, the factorised form of 2x² + 7x + 3 is (x + 3)(2x + 1).

Example: Factorise 3x² - x - 2

1. Find the product ac: a * c = 3 * (-2) = -6

2. Find the factors of ac: We need two numbers that add up to -1 (the coefficient of x) and multiply to -6. These numbers are -3 and 2 (-3 + 2 = -1 and -3 * 2 = -6).

3. Rewrite the middle term: Rewrite -x as -3x + 2x. The quadratic becomes 3x² - 3x + 2x - 2.

4. Factor by grouping: 3x(x - 1) + 2(x - 1)

5. Factor out the common binomial: (x - 1)(3x + 2)

Therefore, the factorised form of 3x² - x - 2 is (x - 1)(3x + 2).

Method 3: Using the Quadratic Formula (For Difficult Quadratics)

While factorisation by inspection and grouping are efficient for many quadratics, some expressions are difficult or impossible to factorise using these methods. In such cases, the quadratic formula provides a reliable alternative. The quadratic formula solves for the roots (values of x that make the quadratic equal to zero) of the equation ax² + bx + c = 0. The roots, often denoted as α and β, are given by:

x = [-b ± √(b² - 4ac)] / 2a

Once you have the roots, you can express the factorised form as:

a(x - α)(x - β)

Example: Factorise 6x² + 7x - 3

1. Use the quadratic formula: a = 6, b = 7, c = -3 x = [-7 ± √(7² - 4 * 6 * -3)] / (2 * 6) x = [-7 ± √(49 + 72)] / 12 x = [-7 ± √121] / 12 x = [-7 ± 11] / 12

2. Find the roots: x₁ = (-7 + 11) / 12 = 4 / 12 = 1/3 x₂ = (-7 - 11) / 12 = -18 / 12 = -3/2

3. Write the factorised form: 6(x - 1/3)(x + 3/2) This can be simplified by multiplying the fractions back into the brackets: 6(3x - 1)/3 * (2x + 3)/2 (3x - 1)(2x + 3)

Therefore, the factorised form of 6x² + 7x - 3 is (3x - 1)(2x + 3).

Perfect Square Trinomials

A perfect square trinomial is a quadratic that can be expressed as the square of a binomial. They have a specific pattern:

(px + q)² = p²x² + 2pqx + q² or (px - q)² = p²x² - 2pqx + q²

Recognising this pattern can significantly simplify factorisation.

Example: Factorise x² + 6x + 9

This is a perfect square trinomial because:

• The first term (x²) is a perfect square (x² = x * x)
• The last term (9) is a perfect square (9 = 3 * 3)
• The middle term (6x) is twice the product of the square roots of the first and last terms (2 * x * 3 = 6x)

Therefore, x² + 6x + 9 = (x + 3)²

Difference of Squares

Another special case is the difference of squares, where a quadratic can be written as the difference between two perfect squares:

p²x² - q² = (px + q)(px - q)

Example: Factorise 4x² - 25

This is a difference of squares because:

• 4x² is a perfect square (2x * 2x)
• 25 is a perfect square (5 * 5)

Therefore, 4x² - 25 = (2x + 5)(2x - 5)

Frequently Asked Questions (FAQ)

Q1: What if I can't find the factors easily?

If you're struggling to find the factors by inspection or grouping, use the quadratic formula. It always works, even if the quadratic is difficult to factorise by other methods.

Q2: Are there other methods for factorising quadratics?

While the methods described above are the most common and widely used, some advanced techniques exist, like completing the square. However, these are usually not necessary for most problems.

Q3: Why is factorising quadratics important?

Factorising quadratics is essential for solving quadratic equations, simplifying algebraic expressions, and understanding more advanced mathematical concepts such as calculus and conic sections.

Conclusion

Factorising quadratics is a crucial skill in algebra. Mastering this skill empowers you to solve a wide range of mathematical problems efficiently. Remember to start by identifying the type of quadratic you're working with. If it's a simple quadratic (a = 1), try factoring by inspection. For general quadratics, factoring by grouping is a reliable method. If you encounter difficulties, don't hesitate to use the quadratic formula, which guarantees a solution. By practicing regularly and understanding the different methods, you will develop the confidence and expertise to tackle any quadratic factorisation problem you encounter. Remember to always check your work by expanding the factorised form to ensure it matches the original quadratic. Happy factoring!