The equation \({x^2} - 3x + {k^2} = 4\) has two distinct real roots. Find the possible values of k .

evidence of rearranged quadratic equation (may be seen in working)     A1

e.g. \({x^2} - 3x + {k^2} - 4 = 0\) , \({k^2} - 4\) 

evidence of discriminant (must be seen explicitly, not in quadratic formula)     (M1)

e.g. \({b^2} - 4ac\) , \(\Delta  = {( - 3)^2} - 4(1)({k^2} - 4)\)

recognizing that discriminant is greater than zero (seen anywhere, including answer)     R1

e.g. \({b^2} - 4ac > 0\) , \(9 + 16 - 4{k^2} > 0\)

correct working (accept equality)     A1

e.g. \(25 - 4{k^2} > 0\) , \(4{k^2} < 25\) , \({k^2} = \frac{{25}}{4}\)

both correct values (even if inequality never seen)     (A1)

e.g. \(\pm \sqrt{{\frac{{25}}{4}}}\) , \( \pm 2.5\)

correct interval     A1     N3

e.g. \( - \frac{5}{2} < k < \frac{5}{2}\) , \( - 2.5 < k < 2.5\)

Note: Do not award the final mark for unfinished values, or for incorrect or reversed inequalities, including \( \le \) , \(k > - 2.5\) , \(k < 2.5\) .

Special cases:

If working shown, and candidates attempt to rearrange the quadratic equation to equal zero, but find an incorrect value of c, award A1M1R1A0A0A0.

If working shown, and candidates do not rearrange the quadratic equation to equal zero, but find \(c = {k^2}\) or \(c = \pm 4\) , award A0M1R1A0A0A0.

[6 marks]

The majority of candidates who attempted to answer this question recognized the need to use the discriminant, however very few were able to answer the question successfully. The majority of candidates did not recognize that the quadratic equation must first be set equal to zero. In addition, many candidates simply set their discriminant equal to zero, instead of setting it greater than zero. Even many of the strongest candidates, who obtained the correct numerical values for k, were unable to write their final answers as a correct interval.

This question is a good example of candidates who reach for familiar methods, without really thinking about what the question is asking them to find. There were many candidates who attempted to solve for x using the quadratic formula or factoring, even though the question did not ask them to solve for x.