# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2017 | Feb-Mar | (P1-9709/12) | Q#1

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Question

Find the set of values of k for which the equation has distinct real roots.

Solution

We are given the equation; Standard form of quadratic equation is; Expression for discriminant of a quadratic equation is; If ; Quadratic equation has two distinct real roots.

If ; Quadratic equation has no real roots.

If ; Quadratic equation has one real root/two equal roots.

We are given that the given equation has distinct real roots, therefore;   To find the set of values of k for which ; we solve the following equation to find critical  values of ;  Now we have two options;      Hence the critical points on the curve for the given condition are 0 & .

Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards  and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the  graph.

We recognize that given curve , is a parabola opening upwards. Therefore conditions for are;  