Solve Quadratic Equations by Factorising
It is concerning that many students shy away from Algebra because of the abstract nature of the topic. But Algebra, (especially Solving Quadratic Equations by Factorising), in an interesting and stimulating concept. If taught well, students can grasp the concept.
Here are attempts on solving quadratic equations by factorising by using the:
- ‘difference of two squares’ (example #1) and
- cross-multiplication technique (example #2).
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.
How to solve a quadratic equation
To solve quadratic equations (equations of the highest power of 2), it is important to factorise the equations first. The key steps are:
- identify the difference between simple and complex quadratic equations
- determine when to use the factoring method and the quadratic formula to solve quadratic equations
- solve simple and complex quadratics equations
Solve simple quadratic expressions (I)
Factorising and solving simple quadratic expressions (and equations) like
x^2 – 4 (difference of 2 squares) and x^2 + 4x + 4 can be done at Grade 10 levels. these quadratic expressions (and equations), introduced in Grade 9 and revised at Grade 10, often paves way for harder questions at 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
x^2 – 4=(x^2 – 2^2 = (x + 2)(x – 2)
(II) Solve the equation: x^2 – 64 = 0
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!
Practice #1: Factories and solve
x^2 – 25 = 0
Comment with your answer
Solve Quadratic Equation by Factorising (II)
Some quadratic equations such as 8x^2 – 16x + 6 =0 can be factorised and then solved.
They are in the form ax^2 + bx + c = 0, where they can assume a – or positive variables like 8, -16 and 6 in the given equation.
In that case, students can use the factor method and, by intelligent guess and check, find the factors that are common. See example #2.
Factorise and solve 8x^2 – 16x + 6 =0
It is a four-step simplified process:
(i) Identify the first and last terms
(ii) find the ‘right factors’
(ii) use cross-multiplication technique to find best combinations;
(iv) factorise, then
(v) solve the equation
Example #2: Solve the quadratic equation 10x^2 + 12x – 12=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
(4x – 2) (2x – 3) = 0
4x – 2 = 0 OR 2x – 3 = 0
4x = 2 OR 2x = 3
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
Facts to know when Solving Quadratic Equations by Factorising
- x^2 – 4 is a quadratic equation with 2 terms – ‘difference of two squares’
- x^2 – 4x + 4 is a quadratic expression with 3 terms, x is a variable and 4 is a constant
- x^2 – 4x + 4 = 0 is a quadratic equation
- Factors of Quadratic Equations have to be equal to ZERO
- ‘Solve’ means to find the value of x
- The two common methods of solving a quadratic equation are by factorising and using quadratics formula. The third method is by graphing the equation and identifying x-intercepts.
- There are always two solutions for a quadratic equation
- The general quadratic equation is given as ax^2 + bx + c = 0