It is concerning that many students shy away from Algebra because of the abstract nature of the topic. But Algebra, (especially Solve Quadratic Equation by Factorising), is an interesting and stimulating concept. If taught well, students can grasp the concept.
Here are attempts at solving quadratic equations by factorising using the:
- ‘difference of two squares’ (example #1) and
- cross-multiplication technique (example #2).
- Another option is to use the Quadratic Formula.
For students, if you get the basics right, you can make connections between the simple and complex algebra (Quadratic Equations) problems, and have lots of fun.
So, let’s have a look at some clear (and orderly) illustrations for Solving Quadratic Equations by Factorisation. You can use the examples to do the practice exercises as you read and understand the samples presented here.
The examples are modelled around the GCSE (UK/Edexcel) and PNG Examination questions and are relevant to Grade 10, 11 and 12 students. Download the Algebra practice questions and get more practice.
Solve quadratic expressions (I)
Factorising and solving simple quadratic expressions (and equations) like x^2 – 4 and x^2 + 4x + 4 can be done at Grade 10 levels. These quadratic expressions (and equations) are introduced in Grade 9 and revised in Grade 10. And later covered in detail in Grades 11 and 12. So a thorough understanding is vital.
Example #1. The difference of 2 squares
a^2+b^2=(a + b)(a-b) (^ mean to the power of)
(I) Factorise the expression x^2 – 4 (difference of two squares)
x^2 – 4=(x^2 – 2^2 = (x + 2)(x – 2)
(II) Solve the equation: x^2 – 64 = 0 (difference of two squares)
x^2 – 64 = 0
x^2 – 8^2 = 0
(x + 8)(x – 8) = 0… difference of two squares
x + 8=0 or x – 8 = 0… for the eqn to equal to 0
So, x = -8 or x = 8…. solution!
Here is a question on the difference of two squares, try it out.
x^2 – 25 = 0
Solve Quadratic Equation by Factorising (II)
Some quadratic equations such as 8x^2 – 16x + 6 =0 can be factorised and solved. This example is in the form ax^2 + bx + c = 0, where they can assume a + or – variable like 8, -16 and 6 in the given equation.
In this example, we will find the factors in two steps: identify factors and factorise, and solve the problem by finding the values of the unknown, x.
STEP 1 – Identify the possible factors
Factorise and solve 8x^2 – 16x + 6 =0
(i) Identify the first and last terms: 8x^2 and + 6
(ii) Find factors by guess-and-check: 8x^2 = 4x * 2x ( * means multiply) and +6 = 3 * 2
(iii) Find the best combinations by cross-multiplication: (this is the tricky step)
STEP 2: Solve the equation
1. (4x – 2) (2x – 3) = 0
2. 4x – 2 = 0 OR 2x – 3 = 0
3. 4x = 2 OR 2x = 3
4. x = 1/2 or x = 2/3
(i) Identify the first and the last terms
(ii) Find numbers (also called multiples) when CROSS multiplied gives the first and the last terms
(iii) Adjust the products of the first and last terms so that they give the MIDDLE term
(iv) On passing step (iii), take the factors ACROSS the top of the cross-multiplication arrangement (step ii) – the equation CAN now be factorised
(v) Now, solve the equation by equating the factors to 0.
Practice #2: Solve the quadratic equation
12x^2 + 20x – 25 = 0
Comment with your answer
How to Solve Quadratic Equations by Factorising
- 1. x^2 – 4 is a quadratic equation with 2 terms – ‘difference of two squares’
- 2. x^2 – 4x + 4 is a quadratic expression with 3 terms, x is a variable and 4 is a constant
- 3. x^2 – 4x + 4 = 0 is a quadratic equation
- 4. Factors of Quadratic Equations have to be equal to ZERO
- 5. ‘Solve’ means to find the value of x
- 6. The two common methods of solving a quadratic equation are factorising and using the quadratic formula. The third method is by graphing the equation and identifying x-intercepts.
- 7. There are always two solutions for a quadratic equation
- 8. The general quadratic equation is given as ax^2 + bx + c = 0