How to Factorise Quadratic Equation
When it comes to algebra, quadratic equations are one of the most important topics. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants. One of the most important skills in solving these equations is understanding how to factorise them. In this article, we will explain step-by-step how to factorise a quadratic equation.
Factorising Quadratic Trinomials
A quadratic trinomial is a quadratic equation that has three terms. To factorise a quadratic trinomial, follow these steps:
- Write the trinomial in the form ax² + bx + c = 0
- Find two numbers that multiply to give you a x c, and add up to give you b
- Replace bx with the two numbers you found in step 2
- Group the first two terms and the last two terms
- Factor out the common factor from each group
- Factor out the common factor from the entire equation
Let’s use an example to illustrate these steps. Suppose we want to factorise the quadratic trinomial x² + 7x + 10. We can write this in the form ax² + bx + c = 0, where a = 1, b = 7, and c = 10. We need to find two numbers that multiply to give us 10, and add up to give us 7. Those numbers are 2 and 5. We can replace 7x with 2x + 5x to get x² + 2x + 5x + 10. We can then group the first two terms and the last two terms to get (x² + 2x) + (5x + 10). We can factor out x from the first group to get x(x + 2), and factor out 5 from the second group to get 5(x + 2). We can then factor out the common factor (x + 2) from the entire equation to get (x + 2)(x + 5) = 0. This means that either x + 2 = 0 or x + 5 = 0. Solving for x, we get x = -2 or x = -5.
Factorising Perfect Square Trinomials
A perfect square trinomial is a quadratic equation of the form a² + 2ab + b², where a and b are constants. To factorise a perfect square trinomial, follow these steps:
- Identify if the quadratic is a perfect square trinomial
- Write the perfect square trinomial in the form (a + b)²
- Expand (a + b)² to get a² + 2ab + b²
Let’s use an example to illustrate these steps. Suppose we want to factorise the perfect square trinomial x² + 6x + 9. We can see that this is a perfect square trinomial because the first and last terms are squares (x² and 9), and the middle term is twice the product of the square roots of the first and last terms (2 * x * 3 = 6). We can write x² + 6x + 9 in the form (x + 3)². We can expand (x + 3)² to get x² + 6x + 9. This means that (x + 3)² = 0, which means that x + 3 = 0. Solving for x, we get x = -3.
Factorising Difference of Squares
A difference of squares is a quadratic equation of the form a² – b², where a and b are constants. To factorise a difference of squares, follow these steps:
- Write the difference of squares in the form (a + b)(a – b)
Let’s use an example to illustrate these steps. Suppose we want to factorise the difference of squares x² – 25. We can write x² – 25 in the form (x + 5)(x – 5). This means that either x + 5 = 0 or x – 5 = 0. Solving for x, we get x = -5 or x = 5.
Conclusion
Factorising quadratic equations is an important skill in algebra, and it can be used to solve a variety of problems. By following the steps outlined in this article, you should now have a good understanding of how to factorise quadratic trinomials, perfect square trinomials, and differences of squares. However, it is important to remember that practice makes perfect, so make sure to do plenty of exercises to master these techniques.
