# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | June | Q#8

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Question

Given that the equation 2qx+ qx – 1 = 0, where q is a constant, has no real roots,

a.   show that q2 + 8q < 0.

b.   Hence find the set of possible values of q.

Solution

a.

We are given; We are given that given equation has no real roots.

For a quadratic equation , the expression for solution is; Where is called discriminant.

If , the equation will have two distinct roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

Since given is a quadratic equation with no real roots, its discriminant must be;   b.

We are required to solve the inequality; 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 -8 & 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 it is an upwards opening parabola. Therefore conditions for are;   