Factorising Quadratics: Step-by-Step Examples with Worksheet
Factorising Quadratics
Quadratic expressions, which are an integral part of algebra, illustrate the nuance and complexity of mathematical formulas.
The process of factorizing quadratics is not only a mathematical operation, but it is a problem-solving tool that finds a variety of applications in scientific, engineering, as well as economic problems.
- Factorising quadratics is not just a mathematical operation; it’s a problem-solving tool with applications in various scientific, engineering, and economic contexts.
In this article, we will discuss:
- What is a Quadratic Expression and Factorising Quadratics?
- How to Factorise Quadratics
Here is one more link to practice a few extra questions: Maths Genie Factorising Quadratics Questions
What is a Quadratic Expression?
- A quadratic expression in math consists in a squared term (x2). It has the form ax2 + bx + c, where:
“a” is the coefficient of the x2 term,
“b” is the coefficient of the x-term;
“c” represents the constant term.
The highest power of a quadratic expression is 2.
If there are powers of x raised higher (like x3 or more), it is not a quadratic expression.
Monic Quadratic Expression (a = 1):
When the coefficient of x2 is 1 (for example x2 – 4x – 12), it will be known as a “monic” quadratic expression.
Non-Monic Quadratic Expression (a ≠ 1):
If the x2 coefficient is not 1 (for example, 3x2 – 4x – 12), it is referred to as a “non-monic” quadratic expression.
What is Factorising Quadratics?
- Factorising quadratic equations is just the reverse process of expanding brackets and is useful in solving quadratic equations.
The expression with the form (x2 + bx + c) can be done by two brackets as (x + d)(x + e).
Quadratic Expressions or Quadratic Equations?
Expression vs. Equation:
A quadratic expression (math phrase including variables, constants, and operations) is a mathematical expression.
A quadratic equation in general may be interpreted as the declaration of equality of two expressions with ‘ax2 + bx + c = 0′ in particular.
Sometimes factoring of the numbers is used in solving quadratic equations by finding the values of x that make the equation correct.
Factorising Quadratics in the Form x2 + bx + c
Using the double brackets, we can factorize the quadratic expression as x2 + bx + c.
The process where factorization is found is the opposite of the operation in which double brackets are expanded. The case when the coefficient right after the x2 term is a = 1, is termed monic quadratic.
How to Factorise Quadratics:
Method #1: Factorising “by inspection”
To factorize a quadratic expression, like x2 + 6x + 9, follow these steps:
Identify coefficients:
The quadratic expression is x2 + bx + c.
In this example, b is 6, and c is 9.
Find two numbers:
Find the pair of numbers that multiply to c (9) and add up to b (6).
- The numbers are 3: because
3 x 3 = 9
and
3 + 3 = 6
Write in brackets:
Place these numbers in the brackets like this.
- The factored form is (x + 3)(x + 3) or (x + 3)2
In fact, through observation, you can see that this expression can be factorized either as (x + 3)(x + 3) or as (x + 3)2.
Method #2: Factorising "by grouping"
Method 2: Factorising “by grouping”
- To factorize a quadratic expression, like x2 – 4x – 12, using the “by grouping” method, follow these steps:
Identify coefficients:
- The quadratic expression is x2 – 4x -12.
The coefficient, b is -4, and the y-intercept, c is -12.
Find two numbers:
Consider a set of numbers that when multiplied together result in c (-12), and when added together will equal b (-4).
The values of -6 and +2 satisfy these conditions.
(-6) x 2 = -12
and
(-6) + 2 = -4
Rewrite the middle term:
Rewrite the middle term of the quadratic expression in terms of the two numbers you have identified while factoring.
x2 – 6x + 2x – 12
Group and factorize:
Split terms into groups by two and then out of the pairs find the common factor.
- Factor out x from the first two terms and 2 from the last two terms:
(x(x – 6) + 2(x – 6))
Factor out common factors:
One can notice that both the terms have a common factor of (x – 6) now.
Factor out (x – 6):
(x – 6)(x + 2).
Hence, you can factor this x2 – 4x – 12 as (x – 6)(x + 2).
Solved Example:
Question 1: Factorize x2 – 13x + 36.
Solution:
- Step #1: Identify coefficients:
The quadratic expression is x2 -13x + 36.
The coefficient, b is -13 and c = 36.
- Step #2: Find two numbers:
Seek a pair of numbers that, when multiplied, equal to c (36) and when added, equal to b (-13).
The numbers are -9 and -4 since
(-9) x (-4) = 36
and
(-9) + (-4) = -13
- Step #3: Revise the middle term as follows:
Express the midterm of the quadratic expression with the two factors determined through the process of factorization
x2 – 9x – 4x + 36
- Step #4: Group and factorize:
Group the terms in pairs and extract the common factor from each pair.
Factor out x from the first two terms and -4 from the last two terms:
(x(x – 9) -4(x – 9))
- Step #5: Factor out common factors:
See that we have a common term for both expressions, that is, (x – 9).
Factor out (x – 9):
(x – 9)(x – 4).
So, by grouping, you can factorize x2 – 13x + 36 as (x – 9)(x – 4).
Practice Questions
Question 1. Factorize x2 + 7x + 12
Answer : ( )( )Question 2: Factorize x2 + 8x + 15
Answer : ( )( )Factorising quadratics in the form ax2 + bx + c
To factorize quadratic expressions in the form ax2 + bx + c, apply the double brackets twice. Factorization is the opposite of expanding double brackets.
When the coefficient in front of the x2 term (a greater than 1) is greater than 1, it’s a non-monic quadratic formula.
How to Factorize Quadratics: ax2 + bx + c (Double Brackets)
- Here’s a step-by-step guide on how to factorize quadratics in the form ax2 + bx + c into double brackets:
- Multiply the coefficients of x2 term (a) and the constant term (c).
- Write out the factor pairs of this product in order
- Look for a pair of factors from step#1 that add up to the coefficient of x term (b).
- Rewrite the original expression, splitting the middle term into the two factors found in step#2
- The order of these factors doesn’t matter; the sign do.
- Split the equation down the middle and fully factorize each half.
- Ensure that the expressions in the brackets are the same.
- Factorize the whole expression by bringing the contents of the bracket to the front and writing the two other terms in the other bracket.
Solved Example:
Question: Factorize 5x2 – 13x – 6
Solution:
- Step #1: Multiply and Identify Factors Pairs
Multiply the coefficients of the x2 term (5) and the constant term (-6):
5 x (-6) = -30
Write out the factor pairs of -30 in order:
(-1,30), (-2, 15), (-3, 10), (-5, 6).
- Step #2: Find Factors for Middle Term:
Look for a pair of factors from step 1 that add up to the coefficient of the x term (-13).
The pair is (-15,2) because (-15) + 2 = -13
- Step #3: Rewrite the Expression:
Rewrite the original expression, splitting the middle term into the two factors found in step #2
5x2 – 15x + 2x – 6
- Step #4: Split and Fully Factorize:
Split the equation down the middle:
5x2 – 15x and 2x – 6.
Fully factorize each half:
5x(x – 3) and 2(x – 3).
- Step #5: Factorise the Whole Expression:
Factorize the whole expression by bringing the contents of the brackets to the front and writing the two other terms in the other bracket.
(5x + 2)(x – 3)
So, the solution to 5x2 -13x -6 by factorizing into double brackets is (5x + 2) (x – 3)
Practice Questions
Question 1. Factorize 2x2 + 9x + 10
Answer : ( )( )Question 2: Factorize 5x2 + 26x + 5
Answer : ( )( )Conclusion
Factorising quadratics is one of the important skills for a deeper understanding of algebraic statements.
Comprehension of factorisation entails a unique mathematical instrument that can be used in numerous disciplines.
This skill is varied from basic to creative mathematics.
Worksheet on Factorising Quadratics
Question 1: Factorize x2 + 10x + 25
Question 2: Factorize x2 + 9x + 14
Question 3: Factorize 2x2 + 17x + 36
Question 4: Factorize 5x2 + 62x + 24
Question 5: Factorize 7x2 + 10x + 3