Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2011  January  Q#8
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Question
The equation x^{2} + (k − 3)x + (3− 2k) = 0, where k is a constant, has two distinct real roots.
a. Show that k satisfies
k^{2 }+ 2k – 3>0
b. Find the set of possible values of k.
Solution
a.
We are given that;
We are given that given equation has two distinct 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 two different 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 ;
Now we have two options;






Hence the critical points on the curve for the given condition are 1 & 3.
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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