CAIE IGCSE Additional Mathematics 2.3 Discriminant Conditions Question Bank

Practise using b²−4ac to determine parameter ranges for real, repeated or non-real roots and tangency.

Exam points

form b²−4ac and compare it with zero for the required root condition
solve the resulting inequality to obtain the permitted parameter interval
set the discriminant to zero when a line is tangent to a quadratic curve

[Maximum number: 1]

The polynomial p is given by $\mathrm{p}(x)=a^{2} x^{3}+2 a x^{2}+a x+2$ , where a is a positive integer. It is given that 2 x+1 is a factor of p(x).

Hence show that the equation p(x)=0 has only one real root.

Discriminant of $4 x^{2}+1$ is 0-16<0 oe and no real roots oe or $4 x^{2}+1=0 \rightarrow 4 x^{2}=-1$ oe and therefore no solution oe

[statement that only solution is $x=-\frac{1}{2}$ ]

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2.3. Use discriminant conditions question 1