Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#7
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Question
where k is a real constant.
a. Find the discriminant of f(x) in terms of k.
b. Show that the discriminant of f (x) can be expressed in the form (k + a)2 + b, where a and b are integers to be found.
c. Show that, for all values of k, the equation f (x) = 0 has real roots.
Solution
a.
We are given;
It is evident that we are given a quadratic equation and we are required to find discriminant of this equation.
For a quadratic equation , the expression for solution is;
Where is called discriminant.
Therefore;
b.
We have found discriminant of given quadratic equation in (a) as;
We have the expression;
We use method of “completing square” to obtain the desired form.
Next we complete the square for the terms which involve .
We have the algebraic formula;
For the given case we can compare the given terms with the formula as below;
Therefore we can deduce that;
Hence we can write;
To complete the square we can add and subtract the deduced value of ;
c.
We are given that f(x)= has real roots.
Since ;
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 we are given only equation has real roots but not that these roots are distinct or otherwise.
Therefore, for this equation;
We have found in (a) that discriminant of this equation is;
Therefore, we need to solve the inequality;
We have also demonstrated in (b) that can be written as;
Hence;
It is evident that for all real values of k the above equation will be;
Hence, the given equation will have real roots for all real values of k.
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