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The coordinates of the point in question are expressed as (4p, p^2), and it is stated that this point resides on the line defined by the equation 2x − 4y + 5 = 0. To determine the values of the constant p that satisfy this condition, one must substitute the coordinates of the point into the line’s equation. This process involves replacing x with 4p and y with p^2, leading to the formulation of an equation that can be solved for p.
Upon substituting these values into the line equation, we arrive at the expression 2(4p) − 4(p^2) + 5 = 0. Simplifying this equation yields 8p − 4p^2 + 5 = 0. Rearranging the terms results in a standard quadratic equation of the form -4p^2 + 8p + 5 = 0. To facilitate the solution, one may multiply through by -1, transforming the equation into 4p^2 – 8p – 5 = 0, which is now ready for application of the quadratic formula or factoring techniques.
By employing the quadratic formula, p can be calculated as p = [8 ± √(64 + 80)] / 8, which simplifies to p = [8 ± √144] / 8. This results in two potential solutions for p, specifically p = 3 and p = -\(\frac{1}{2}\). Thus, the two possible values of the constant p that allow the point (4p, p^2) to lie on the specified line are 3 and -\(\frac{1}{2}\). For those seeking further assistance in understanding this process, a mathematics tutor can provide valuable guidance and clarification.
