Algebra - Solving equations and inequalities: solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph (Higher tier)

The idea

Higher tier only. On top of factorising you need two methods that work on any quadratic, and the habit of tidying up first: x(x + 4) = 12 must become x² + 4x − 12 = 0 before any method applies. Completing the square rewrites x² + 6x + 2 = 0 as (x + 3)² − 7 = 0, so (x + 3)² = 7 and x = −3 ± √7. The formula x = (−b ± √(b² − 4ac)) / (2a) runs off the coefficients alone and is the safe choice when nothing factorises neatly. Graphically, plot y = x² + 6x + 2 and read the x-coordinates where the curve crosses the horizontal axis as approximate roots. Read the wording for the accuracy wanted: an exact or surd answer keeps the root in place, while a rounded answer means working out the two roots separately, once with the plus and once with the minus.

What earns the marks

• Rearranging to a quadratic with zero on one side, including expanding brackets or clearing a fraction first.
• Substituting the values of a, b and c into the formula with their signs kept, even if the arithmetic afterwards is wrong.
• The correct value under the root sign before it is simplified.
• A correct completed square form, and a further mark for the solutions that follow from it.
• On a graphical question, plotting the curve and reading the x-coordinates of the intersections with the horizontal axis as approximate solutions.

Common mistakes and how to put them right

• Forgetting that the formula begins with −b, so a positive coefficient is substituted without its sign changing.

• Taking a to be the coefficient of x rather than the coefficient of x².

• Halving the coefficient of x when completing the square but not subtracting the square of that half, so x² + 6x is written as (x + 3)² instead of (x + 3)² − 9.

• Dividing only the first part of the formula by 2a and leaving the root undivided.

• Assuming a negative value under the root sign must be a slip, when it can legitimately mean the quadratic has no real roots.

Exam tips

• Write down a, b and c with their signs before substituting, and put brackets round any negative value.
• A negative number under the root sign may be a sign slip, but it can also be a valid no-real-roots result, so check the coefficients and then interpret the outcome.
• Completing the square also hands you the turning point of the curve, which a later part of the same question often asks for.