Factorising Quadratics
Why is factorising quadratics useful?
- Factorising gives roots (zeroes or solutions) of a quadratic
- It gives the x-intercepts when drawing the graph
How do I factorise a monic quadratic of the form x2 + bx + c?
- You might be able to spot the factors by inspection
- Especially if c is a prime number
- Otherwise find two numbers m and n ..
- A sum equal to b
- A product equal to c
- A sum equal to b
- Rewrite bx as mx + nx
- Use this to factorise x2 + mx + nx + c
- A shortcut is to write:
How do I factorise a non-monic quadratic of the form ax2 + bx + c?
- If a, b & c have a common factor then first factorise that out to leave a quadratic with coefficients that have no common factors
- You might be able to spot the factors by inspection
- Especially if a and/or c are prime numbers
- Otherwise find two numbers m and n ..
- A sum equal to b
- A product equal to ac
- A sum equal to b
- Rewrite bx as mx + nx
- Use this to factorise ax2 + mx + nx + c
- A shortcut is to write:
- Then factorise common factors from numerator to cancel with the a on the denominator
How do I use the difference of two squares to factorise a quadratic of the form A2x2 - C2?
- The difference of two squares can be used when...
- There is no x term
- The constant term is a negative
- Square root the two terms A2x2 & C2
- The two factors are the sum of square roots and the difference of the square roots
- A shortcut is to write:
Exam Tip
- You can deduce the factors of a quadratic function by using your GDC to find the solutions of a quadratic equation
- Using your GDC, the quadratic equation has solutions and
- Therefore the factors would be and
- i.e.
Worked Example
Factorise fully:
a)
.
b)
.
c)
.
Completing the Square
Why is completing the square for quadratics useful?
- Completing the square gives the maximum/minimum of a quadratic function
- This can be used to define the range of the function
- It gives the vertex when drawing the graph
- It can be used to solve quadratic equations
- It can be used to derive the quadratic formula
How do I complete the square for a monic quadratic of the form x2 + bx + c?
- Half the value of b and write
- This is because
- Subtract the unwanted term and add on the constant c
How do I complete the square for a non-monic quadratic of the form ax2 + bx + c?
- Factorise out the a from the terms involving x
- Leaving the c alone will avoid working with lots of fractions
- Complete the square on the quadratic term
- Half and write
- This is because
- Subtract the unwanted term
- Half and write
- Multiply by a and add the constant c
Exam Tip
- Some questions may not use the phrase "completing the square" so ensure you can recognise a quadratic expression or equation written in this form
Worked Example
Complete the square:
a)
.
b)
.