This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Determine the roots of 3x^2 - 5x - 7 = 0 from the graph. The roots of the equation 3x^2 - 5x - 7 = 0 are the x-intercepts of the graph y = 3x^2 - 5x - 7, which are the points where the curve crosses the x-axis (where y=0). From the graph, the curve crosses the x-axis at approximately x = -0.9 and x = 2.6. a) (i) The roots are x = -0.9 and x = 2.6. Step 2: Determine the minimum value of y from the graph. The minimum value of y for this parabola (which opens upwards) is at its vertex, the lowest point on the curve. From the graph, the lowest point on the curve is at y = -9. a) (ii) The minimum value of y is y = -9. Step 3: Calculate the gradient of the curve at point x = 2. To find the gradient of the curve at x=2 from the graph, we need to find the gradient of the tangent line to the curve at x=2. The image provides a calculation using two points on what appears to be a tangent line or a secant line approximating the tangent. The points used are (x_1, y_1) = (1.30, -10) and (x_2, y_2) = (3, 1.5). The formula for the gradient (slope) is: m = ( y)/( x) = (y_2 - y_1)/(x_2 - x_1) Substitute the given points: m = (1.5 - (-10))/(3 - 1.30) m = (1.5 + 10)/(1.70) m = (11.5)/(1.70) m ≈ 6.7647 Rounding to one decimal place, the gradient is 6.8. a) (iii) The gradient of the curve at point x=2 is ≈ 6.8.