213. Factorising Monic Quadratics

Learning Intentions

• Factorise monic quadratic equations with integer roots.
• Apply the null factor law to Solve solutions.
• Check solutions by substitution.

Pre-requisite Summary

• Expand brackets using the distributive law.
• Recognise common factors in algebraic expressions.
• Understand that a quadratic equation has degree .
• Recall how to solve simple linear equations.
• Understand that solutions satisfy the equation.
• Substitute values into algebraic expressions correctly.
• Apply order of operations accurately.

Worked Examples

Worked Example 1

Factorise:

Worked Example 2

Factorise:

Then solve:

using the null factor law.

Worked Example 3

Solve:

using factorisation.

Worked Example 4

Solve:

Then check each solution by substitution.

Worked Example 5

Solve:

by factorising and applying the null factor law.

Problems

Problem 1

Factorise:

Problem 2

Factorise:

Then solve:

using the null factor law.

Problem 3

Solve:

using factorisation.

Problem 4

Solve:

Then check each solution by substitution.

Problem 5

Solve:

by factorising and applying the null factor law.

Exercises

Understanding and Fluency

Exercise 1

Factorise:

Exercise 2

Factorise:

Exercise 3

Factorise:

Exercise 4

Solve:

Exercise 5

Solve:

Exercise 6

Solve:

Exercise 7

Solve:

Exercise 8

Solve:

Exercise 9

Check whether is a solution to:

Exercise 10

Check whether is a solution to:

Reasoning

Exercise 11

Explain why:

has two possible solutions.

Exercise 12

A student states that:

Explain why the factorisation is incorrect.

Exercise 13

Describe how factorisation can help solve quadratic equations more efficiently.

Exercise 14

Explain why solutions should be checked by substitution.

Problem-solving

Exercise 15

The area of a rectangle is modelled by:

a) Factorise the expression

b) Determine possible side lengths

Exercise 16

A projectile follows the equation:

a) Solve the equation

b) Check the solutions by substitution

c) Interpret the meaning of the solutions

Exercise 17

A garden has dimensions represented by:

a) Solve the equation

b) Check the solutions

c) Determine which solution is reasonable in context

Potential Misunderstandings

• Forgetting that the quadratic must equal zero before factorising to solve.
• Choosing factor pairs that add incorrectly.
• Confusing the signs of factors.
• Applying the null factor law before fully factorising.
• Forgetting that each factor can equal zero.
• Making arithmetic errors when substituting solutions back in.
• Assuming only one solution exists.
• Expanding factors incorrectly when checking answers.
• Forgetting to reject unreasonable solutions in real-world contexts.

Next: 214. Connecting Graphs and Solutions of Quadratics