Factorising quadratics where a ≠ 1: box method

Task goals

• Draw a 2-by-2 box with ax^2 in the top-left cell and c in the bottom-right cell (opposite corners, never sharing a row or column).
• Find the pair of numbers that multiplies to ac and adds to b, and write them as the other two cells of the box.
• Read the two brackets off the highest common factor of each row (written on the left) and each column (written on top), including the sign when a row or column HCF is negative.
• Check the split is correct using the diagonal-product rule (both diagonals multiply to ac x^2), and check the finished factorisation by expanding it back out.
• Use the box method for reasoning tasks: spot and correct a wrong factorisation, find an unknown coefficient from a partially given factorised form, or find one side of a rectangle given its area and the other side.