Solving Quadratic Equations by Factorising: A GCSE Mathematics Guide
Introduction
Quadratic equations, a crucial topic in GCSE Mathematics, represent real-world situations involving quadratic relationships. Solving them by factorisation is a fundamental skill that unlocks complex mathematical problems.
Main Content
What is an Equation?
An equation is a mathematical statement of equality between two expressions. A quadratic equation is a polynomial equation of degree two, represented in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
Factorisation
Factorisation is the process of breaking down an algebraic expression into its constituent factors. The goal is to find two factors whose product equals the original expression. To factorise quadratics, we look for two numbers whose product is c and whose sum is b.
Step-by-Step Method
1. Move the constant to the right-hand side: ax² + bx = -c
2. Factorise the left-hand side: (ax + m)(ax + n) = -c
3. Equate the constant terms: mn = -c
4. Find two numbers that satisfy mn = -c and m + n = b: (m, n) = [(-b ± √(b² - 4ac)) / 2a]
5. Substitute m and n into the factors: (ax + m)(ax + n) = 0
6. Solve each factor for x: x = -m/a, x = -n/a
Practice Problems
1. Solve x² - 5x + 6 = 0.
- Factorise: (x 2)(x 3) = 0
- Solve: x = 2, x = 3
2. Solve 2x² + 5x - 3 = 0.
- Factorise: (2x 1)(x + 3) = 0
- Solve: x = 1/2, x = 3
Common Mistakes
- Incorrect factorisation: Ensure that the product of the factors equals the constant term and their sum equals the coefficient of x.
- Dividing by x: Avoid dividing by x when one factor is (ax + m) since it may introduce extraneous solutions.
- Forgetting to check solutions: Plug the solutions back into the original equation to ensure they satisfy it.
Conclusion
Solving quadratic equations by factorisation is a powerful technique in GCSE Mathematics. By understanding the concept of factorisation and following the step-by-step method, students can confidently tackle real-world problems and excel in their exams. Remember to practice regularly and consult your teacher or tutor for any clarifications.
FAQ
- What is the discriminant of a quadratic equation? The discriminant (b² 4ac) determines the number and type of solutions.
- How do I solve quadratic equations with no real solutions? If b² 4ac < 0, the equation has complex solutions that cannot be represented as real numbers.
- Can I use other methods to solve quadratic equations? Yes, other methods include completing the square and using the quadratic formula.