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To ascertain the values of k for which the graphs represented by the equations y = 2x^2 − kx and y = x^2 − k intersect, one must first set the two equations equal to each other. This leads to the equation 2x^2 − kx = x^2 − k. By rearranging this equation, we can consolidate the terms to form a quadratic equation in the standard form. Specifically, this results in the equation x^2 − kx + k = 0, which is essential for determining the conditions under which the two graphs intersect.
Next, the intersection points of the graphs can be analyzed by examining the discriminant of the resulting quadratic equation. The discriminant, given by the expression b^2 − 4ac, plays a crucial role in identifying the nature of the roots. For the graphs to intersect at real points, the discriminant must be non-negative. Therefore, we need to ensure that the condition k^2 − 4(1)(k) ≥ 0 holds true. This inequality can be simplified and factored to yield the critical values of k that will allow for the intersection of the two graphs.
In the context of A Level Maths Easter Revision, understanding the implications of the discriminant and the conditions for intersection is vital. By solving the inequality derived from the discriminant, one can determine the specific range of k values that facilitate the intersection of the two quadratic functions. This analytical approach not only reinforces the concepts of quadratic equations but also enhances problem-solving skills essential for success in advanced mathematics.
