Skip to content# Factorising Quadratics: Step-by-Step Examples with Worksheet

## What is a Quadratic Expression?

**Monic Quadratic Expression (a = 1):****Non-Monic Quadratic Expression (a ≠ 1):**## What is Factorising Quadratics?

## Quadratic Expressions or Quadratic Equations?

## Factorising Quadratics in the Form x^{2} + bx + c

## How to Factorise Quadratics:

## Method #2: Factorising "by grouping"

### Practice Questions

**Solution:**

## Factorising quadratics in the form ax^{2} + bx + c

## How to Factorize Quadratics: ax^{2} + bx + c (Double Brackets)

**Step #1: Multiply and Identify Factor Pairs:****Step #3: Rewrite the Expression:****Step #4: Split and Fully Factorise:****Step #5: Factorise the Whole Expression:**### Practice Questions

**Solution:**

## Conclusion

## Worksheet on Factorising Quadratics

**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.

**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

- A quadratic expression in math consists in a squared term (x
^{2}). It has the form**ax**where:^{2}+ bx + c,

**“a”** is the coefficient of the x^{2} 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 x**, it is not a quadratic expression.^{3}or more)

- When the coefficient of x
^{2}is 1 (for example x^{2}– 4x – 12), it will be known as a**“monic”**quadratic expression.

- If the x
^{2}coefficient is not 1 (for example, 3x^{2}– 4x – 12), it is referred to as a**“non-monic”**quadratic expression.

- Factorising quadratic equations is just the reverse process of expanding brackets and is useful in solving quadratic equations.
The expression with the form

**(x**can be done by two brackets as^{2}+ bx + c)**(x + d)(x + e)**.

**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
**‘ax**in particular.^{2}+ bx + c = 0′ - Sometimes factoring of the numbers is used in solving quadratic equations by finding the values of x that make the equation correct.

- Using the double brackets, we can factorize the quadratic expression as
**x**^{2}+ 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
**x**term is^{2}**a = 1,**is termed monic quadratic.

**Method #1: Factorising “by inspection”**

- To factorize a quadratic expression, like x
^{2}+ 6x + 9, follow these steps:

- To factorize a quadratic expression, like x

**Identify coefficients:**

- The quadratic expression is
**x**^{2}+ bx + c.

- The quadratic expression is

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”**

- To factorize a quadratic expression, like x
^{2}– 4x – 12, using the “by grouping” method, follow these steps:

**Identify coefficients:**

- The quadratic expression is
**x**^{2}– 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.

**x ^{2} – 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 x^{2} – 4x – 12 as **(x – 6)(x + 2).**

** **

** **

**Solved Example:**

**Question 1: Factorize x ^{2} – 13x + 36.**

Solution:

**Step #1:****Identify coefficients:**

The quadratic expression is **x ^{2} -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

**x ^{2} – 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 x^{2} – 13x + 36 as **(x – 9)(x – 4).**

Question 1. Factorize **x ^{2} + 7x + 12**

**Step #1:****Identify coefficients:**

The quadratic expression is **x ^{2} + 7x + 12**

The coefficient, **b is 7**, and **c is 12**.

**Step #2:****Find two numbers:**

Look for a pair of numbers that multiply to $$c (12) and add up to (7).

The numbers are 3 and 4 because

**3 x 4 = 12 and 3 + 4 = 7**

**Step #3: Rewrite the middle term:**

Rewrite the middle term of the quadratic expression using the two numbers found.

**x ^{2} + 3x + 4x + 12**

**Step #4: Group and factorize:**

Group the terms into pairs and factor out the common factor from each pair.

Factor out x from the first two terms and 4 from the last two terms:

**(x(x + 3) +4(x + 3))**

**Step #5: Factor out common factors:**

Notice that both terms now have a common factor of (x + 3).

Factor out (x + 3):

**(x + 3)(x + 4).**

So, by grouping, you can factorize x^{2} + 7x + 12 as **(x + 3)(x + 4).**

Question 2: Factorize **x ^{2} + 8x + 15**

**Solution:**

**Step #1:****Identify coefficients:**

The quadratic expression is **x ^{2} + 8x + 15**

The coefficient, **b is 8**, and **c is 15**.

**Step #2:****Find two numbers:**

Look for a pair of numbers that multiply to $$c (15) and add up to (8).

The numbers are 3 and 5 because

**3 x 5 = 15 and 3 + 5 = 8**

**Step #3: Rewrite the middle term:**

Rewrite the middle term of the quadratic expression using the two numbers found.

**x ^{2} + 3x + 5x + 15**

**Step #4: Group and factorize:**

Group the terms into pairs and factor out the common factor from each pair.

Factor out x from the first two terms and 5 from the last two terms:

**(x(x + 3) +5(x + 3))**

**Step #5: Factor out common factors:**

Notice that both terms now have a common factor of (x + 3).

Factor out (x + 3):

**(x + 3)(x + 5).**

So, by grouping, you can factorize x^{2} + 8x + 15 as **(x + 3)(x + 5).**

- To factorize quadratic expressions in the form
**ax**apply the double brackets twice. Factorization is the opposite of expanding double brackets.^{2}+ bx + c, - When the coefficient in front of the
**x**^{2}term (a greater than 1) is greater than 1, it’s a non-monic quadratic formula.

- Here’s a step-by-step guide on how to factorize quadratics in the form
**ax**into double brackets:^{2}+ bx + c

- Multiply the coefficients of x
^{2}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 5x ^{2} – 13x – 6**

Solution:

**Step #1:****Multiply and Identify Factors Pairs**Multiply the coefficients of the x^{2}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**5x**^{2}– 15x + 2x – 6

**Step #4:****Split and Fully Factorize:**Split the equation down the middle:

**5x ^{2} – 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 5x

^{2}-13x -6 by factorizing into double brackets is (5x + 2) (x – 3)

Question 1. Factorize **2x ^{2} + 9x + 10**

**Step #1: Identify Coefficients:**

The quadratic expression is **2x ^{2} + 9x + 10.**

The coefficient of x^{2} (a) is 2, the coefficient of x (b) is 9, and the constant term (c) is 10.

**Step #2: Find two numbers:**

Multiply the coefficient of x^{2} by the constant term:

**2 x 10 = 20**

Identify factor pairs of 20 that add up to the coefficient of x(9).

The pair is (5, 4) because 5 x 4 = 20 and 5 + 4 = 9

**Step #3: Rewrite the middle term:**

Rewrite the middle term using the factor pairs:

**2x ^{2} + 5x + 4x + 10**

**Step #4: Group and factorize:**

Group the terms in pairs and extract the common factor from each pair:

**x(2x + 5) + 2(2x + 5).**

**Step #5: Factor out common factors:**

Factor out the common factor (2x + 5)

**(2x + 5) (x + 2)**

So, the quadratic expression 2x^{2} + 9x + 10 factors into (2x + 5)(x + 2).

Question 2: Factorize **5x ^{2} + 26x + 5**

**Solution:**

**Step #1: Identify Coefficients:**

The quadratic expression is **5x ^{2} + 26x + 5.**

The coefficient of x^{2} (a) is 5, the coefficient of x (b) is 26, and the constant term (c) is 5.

**Step #2: Find two numbers:**

Multiply the coefficient of x^{2} by the constant term:

**5 x 5 = 25**

Identify factor pairs of 25 that add up to the coefficient of x(26).

The pair is (1, 25) because 1 x 25 = 25 and 1 + 25 = 26

**Step #3: Rewrite the middle term:**

Rewrite the middle term using the factor pairs:

**5x ^{2} + x + 25x + 5**

**Step #4: Group and factorize:**

Group the terms in pairs and extract the common factor from each pair:

**x(5x + 1) + 5(5x + 1).**

**Step #5: Factor out common factors:**

Factor out the common factor (5x + 1)

**(5x + 1) (x + 5)**

So, the solution of 5x^{2} + 26x + 5 by factorizing is **(5x + 1)(x + 5)**

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.

Question 1: Factorize **x ^{2} + 10x + 25**

Question 2: Factorize **x ^{2} + 9x + 14**

Question 3: Factorize **2x ^{2} + 17x + 36**

Question 4: Factorize **5x ^{2} + 62x + 24**

Question 5: Factorize **7x ^{2} + 10x + 3**

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