Edexcel IAL Mathematics P1.1.4 discriminant of a quadratic

Most direct questions turn a condition into a quadratic first, then use b² − 4ac with the correct inequality for no roots, one root or intersections.

Exam points

Use b² − 4ac < 0, = 0 or ≥ 0 to match roots, tangency or intersections.
Form a quadratic from a line-curve condition before applying the discriminant.
Solve the resulting inequality in k or m, keeping strict and non-strict signs correct.

[Maximum number: 5]

Find the range of possible values of k for which the curve with equation

y=-\frac{k}{x}+k \quad k>0 \quad x \neq 0

does not touch or intersect the line with equation y=3 x+4

For an intersection of the line and curve:
$3x+4=-\frac{k}{x}+k$ $3x^2+(4-k)x+k=0.$
For two distinct intersections, the discriminant is positive:
$(4-k)^2-12k>0$ $k^2-20k+16>0.$
The critical values are
$k=10\pm2\sqrt{21}.$
Hence the required range is
$10-2\sqrt{21}<k<10+2\sqrt{21}.$
M1 dM1 A1

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P1.1.4 - Discriminant of a quadratic question 1