Factorising & Completing the Square (DP IB Analysis & Approaches (AA)): Revision Note

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?

• A monic quadratic is a quadratic where the coefficient of the x² term is 1

• 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

    • 

• 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?

• A non-monic quadratic is a quadratic where the coefficient of the x² term is not equal to 1

• 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

    • 

• 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  and 

• The two factors are the sum of square roots and the difference of the square roots

• A shortcut is to write:

  • 

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

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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

• Multiply by a and add the constant c

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