Integrate the gradient function to find the equation of the curve.

To find the value of p, we first need to find the equation of the curve by integrating its gradient function. Step 1: Integrate the gradient function to find the equation of the curve. The gradient of the curve is given by (dy)/(dx) = 2px + 3. To find the equation of the curve, y(x), we integrate the gradient with respect to x: y = (2px + 3)\,dx y = p(x^2)/(2/2) + 3x + C y = px^2 + 3x + C where C is the constant of integration. Step 2: Use the first point (-2, 0) to form an equation. The curve passes through the point (-2, 0), so substitute x=-2 and y=0 into the curve equation: 0 = p(-2)^2 + 3(-2) + C 0 = 4p - 6 + C C = 6 - 4p (Equation 1) Step 3: Use the second point ((1)/(2), 0) to form another equation. Assumption: The second point is ((1)/(2), 0). The handwritten text appears to show (1)/(x), but this would make the x-coordinate dependent on x, which is not standard for a fixed point. We proceed with the interpretation of ((1)/(2), 0). Substitute x=(1)/(2) and y=0 into the curve equation: 0 = p((1)/(2))^2 + 3((1)/(2)) + C 0 = p((1)/(4)) + (3)/(2) + C 0 = (p)/(4) + (3)/(2) + C (Equation 2) Step 4: Solve the system of equations for p. Substitute Equation 1 (C = 6 - 4p) into Equation 2: 0 = (p)/(4) + (3)/(2) + (6 - 4p) Combine the constant terms and p terms: 0 = (p)/(4) - 4p + (3)/(2) + 6 0 = (p)/(4) - (16p)/(4) + (3)/(2) + (12)/(2) 0 = -(15p)/(4) + (15)/(2) To solve for p, move the term with p to the other side: (15p)/(4) = (15)/(2) Multiply both sides by 4: 15p = (15)/(2) × 4 15p = 15 × 2 15p = 30 Divide by 15: p = (30)/(15) p = 2 The value of p is 2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.