Factorising is one of the key methods for solving quadratic equations. In this guide, we’ll dive deep into the concept of factorisation, specifically focusing on quadratic equations, to make it easy to understand and apply.
What Is a Quadratic Equation?
A quadratic equation is an equation of the form
ax^2 + bx + c = 0, where
c are constants, and
x is the variable we aim to solve for.
What Is Factorising?
Factorising means to rewrite an equation or expression as a product of its factors. In the context of quadratic equations, factorising involves converting the equation into a form
(x - p)(x - q) = 0, where
q are constants. Solving for
x then becomes more straightforward.
Methods of Factorising
There are several methods to factorise a quadratic equation, but we’ll focus on the most common ones:
1. Common Factor Method
- Find the greatest common factor (GCF) of all the terms in the equation.
- Factor out the GCF.
2x^2 + 4x = 0 can be factorised as
2x(x + 2) = 0.
2. Perfect Square Trinomial Method
- Check if the equation is a perfect square trinomial.
- If it is, rewrite it as
(x ± p)^2 = 0.
x^2 + 4x + 4 = 0 can be factorised as
(x + 2)^2 = 0.
3. Quadratic Formula Method
- Use the quadratic formula
x = (-b ± sqrt(b^2 - 4ac)) / (2a) to find the roots.
- Rewrite the equation as
(x - p)(x - q) = 0.
x^2 - 3x + 2 = 0,
x = 1, 2, so it can be factorised as
(x - 1)(x - 2) = 0.
4. Regular Quadratic Trinomials Method
This method is particularly useful for trinomials that can be easily factorised. Here’s how you can use this method:
- Identify the Factor Pairs: Look at the constant term (often represented as ‘c’) in your quadratic equation. Write down all possible pairs of factors that multiply to give you this number.
- Add to Find ‘b’: Find a pair of factors that, when added together, give you the coefficient of the middle term (represented as ‘b’). These numbers should also multiply together to give you the last number (again, ‘c’).
- Set Up Your Brackets: Write down two empty sets of brackets and place the variable (usually ‘x’) at the start of each one.
- Insert the Factors: Place one of the factors you identified earlier in the first set of brackets, and the other factor in the second set. Keep in mind that the signs of the factors are crucial.
Consider the equation
x^2 + 5x + 6 = 0.
- Factor pairs of 6: These would be 1 and 6, or 2 and 3.
- Add to find ‘b’: We find that 2 + 3 equals 5, which is our middle term.
- Brackets: We set it up like this: (x )(x ) = 0.
- Insert Factors: Finally, we can write it as (x + 2)(x + 3) = 0.
And there you have it! You’ve successfully factorised the equation using the Regular Quadratic Trinomials Method. This method is generally quick and efficient, especially if you can quickly identify the factor pairs for the constant term.
5. Completing The Square Method
This method involves turning a quadratic equation into a perfect square trinomial, thus simplifying the solving process.
Solving Quadratic Equations by Factorising
Once the equation is factorised, solving for
x is simple:
- Set each factor equal to zero:
(x - p) = 0 and
(x - q) = 0.
- Solve for
x = p and
x = q.
Factorising quadratic equations can seem daunting at first, but it’s a systematic process that gets easier with practice. Knowing how to properly factorise can not only help in solving equations but also aids in understanding the characteristics of different quadratic equations.