Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2015  June  Q#5
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Question
The equation
, where p is a constant
has no real roots.
a. Show that p satisfies
p^{2 }– 6p +1 > 0
b. Hence, find the set of possible values of p.
Solution
a.
We are given that;
We are given that given equation has no real solutions of x (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 solutions of x (roots), its discriminant must be;
b.
We are required to solve the inequality;
We solve the following equation to find critical values of ;
For a quadratic equation , the expression for solution is;
Therefore;
Now we have two options;



Hence the critical points on the curve for the given condition are & .
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;
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