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Let f(x) = x^2 + (k + 3)x + k where k is a real constant - Edexcel - A-Level Maths Pure - Question 9 - 2011 - Paper 1
Question 9
Let f(x) = x^2 + (k + 3)x + k 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 b... show full transcript
Worked Solution & Example Answer:Let f(x) = x^2 + (k + 3)x + k where k is a real constant - Edexcel - A-Level Maths Pure - Question 9 - 2011 - Paper 1
Step 1
Find the discriminant of f(x) in terms of k.
Answer
The discriminant of a quadratic function of the form is given by the formula . For the function
f(x) = x^2 + (k + 3)x + k:\n\nHere, a = 1, b = k + 3, and c = k.\n\nSubstituting these values into the formula gives us:
Expanding this results in:
This simplifies to:
Step 2
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.
Answer
We have already derived the discriminant as
We can manipulate this expression to fit the required form.
\nNotice that we can complete the square for the quadratic term:
Thus, we can state that:
a = 1 and b = 8.
Step 3
Show that, for all values of k, the equation f(x) = 0 has real roots.
Answer
In order for the quadratic equation f(x) = 0 to have real roots, the discriminant must be non-negative, i.e., .
We previously established that:
Here, is always non-negative since it is a square term, and adding 8 ensures that
$$D \geq 8 > 0.$
Therefore, for all values of k, the equation f(x) = 0 will always have real roots.