If the equations \(x^2 - 5x + 6 = 0\) and \(x + px + 6 = 0\) have the same roots, find the value of p.

The question presents two quadratic equations: x^2 - 5x + 6 = 0 and x + px + 6 = 0. Both equations are said to have the same roots. To find the value of p, we need to determine when two quadratic equations have the same roots. For two quadratic equations, ax^2 + bx + c = 0 and dx^2 + ex + f = 0, to have the same roots, the following conditions must be met: 1. The discriminant of both equations must be zero. 2. The ratios of the corresponding coefficients must be equal. Let's apply these conditions to the given equations: Equation 1: x^2 - 5x + 6 = 0 Equation 2: x + px + 6 = 0 1. Discriminant condition: The discriminant of Equation 1 is given by Δ1 = b1^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1. The discriminant of Equation 2 is given by Δ2 = e^2 - 4df = p^2 - 4(1)(6) = p^2 - 24. Since both equations have the same roots, their discriminants must be equal: Δ1 = Δ2 1 = p^2 - 24 p^2 = 25 p = ±5 2. Ratio of coefficients condition: Comparing the coefficients of the corresponding terms: For Equation 1: a1 = 1, b1 = -5, c1 = 6 For Equation 2: a2 = 1, b2 = p, c2 = 6 The ratio of the coefficients must be equal: a1/a2 = b1/b2 = c1/c2 From a1/a2 = 1/1 = 1 and c1/c2 = 6/6 = 1, we can conclude that: b1/b2 = -5/p = 1 -5/p = 1 p = -5 Therefore, the value of p that satisfies both conditions and makes the equations have the same roots is p = -5.