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D2. Elementary algebra : factorising quadratic functions. Algebraic expressions like (x+1) or (x+2) are called linear. Expressions like these can be multiplied together. This is often refered to as multiplying out brackets, for example, (x + 1)(x + 2)= x2 + 3x + 2 The resulting expression is called a quadratic function. All these functions are quadratic functions: x2 + 7x - 11, x2 - 12x , x2 + 14, 6x2 + x - 1, 5 - x2 Generally, quadratic functions have the form: ax2 + bx + c, where a, b and c are numbers and a ¹ 0 The reverse of multiplying out brackets, ie writing x2 + 3x + 2=(x + 1)(x + 2) is called factorising. We will look at how to factorise quadratic functions when this can be done "by inspection". This means you should be able to do it by a combination of mental arithmetic and trial and improvement. Example: factorise x2 + 4x + 3 Step 1: Write down some "empty" brackets x2 + 4x + 3=  (      )(      ) Step 2: The first terms in each bracket must both be x:   (x    )(x    ) Step 3: The last terms in each bracket must multiply to give 3. Try* 3 and 1   (x + 3)(x + 1) Step 4: Check that you get the correct result when you multiply out the brackets:     Sometimes one or both of the numbers involved may be negative: Example: factorise x2 - 2x - 3 Step 1: Write down some "empty" brackets x2 - 2x - 3=  (      )(      ) Step 2: The first terms in each bracket must both be x:   (x    )(x    ) Step 3: The last terms in each bracket must multiply to give -3. Try* 1 and -3   (x + 1)(x - 3) Step 4: Check that you get the correct result when you multiply out the brackets:     Try*: Your first attempt may not be correct, if so try again with two other numbers that multiply to give the required value. Practise factorising quadratic functions. Press the "Factorise" button to get started, then replace the two question marks with numbers to make the bottom equation match the top equation. x2 + x +   x x x2 x x = x2+ x +