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: Complete Table 3 for the equation y = x^3 + 4x^2 + x - 3 For x = -2: y = (-2)^3 + 4(-2)^2 + (-2) - 3 y = -8 + 4(4) - 2 - 3 y = -8 + 16 - 2 - 3 y = 8 - 2 - 3 y = 6 - 3 y = 3 For x = 2: y = (2)^3 + 4(2)^2 + (2) - 3 y = 8 + 4(4) + 2 - 3 y = 8 + 16 + 2 - 3 y = 24 + 2 - 3 y = 26 - 3 y = 23 Completed Table 3: c|c|c|c|c|c|c|c X & -4 & -3 & -2 & -1 & 0 & 1 & 2 \\ Y & -7 & 3 & 3 & -1 & -3 & 3 & 23 \\ Step 2: Draw the graph of y = x^3 + 4x^2 + x - 3 Plot the points from the completed Table 3: (-4, -7), (-3, 3), (-2, 3), (-1, -1), (0, -3), (1, 3), (2, 23). Use the given scale: 2 cm represents 1 unit on the x-axis and 2 cm represents 5 units on the y-axis. Draw a smooth curve connecting these points to represent the graph of y = x^3 + 4x^2 + x - 3. Step 3: Use the graph to solve the equation x^3 + 4x^2 + (3)/(2)x - 4 = 0 To solve x^3 + 4x^2 + (3)/(2)x - 4 = 0 using the graph of y = x^3 + 4x^2 + x - 3, we need to rewrite the equation in terms of y. Given equation: x^3 + 4x^2 + (3)/(2)x - 4 = 0 We know y = x^3 + 4x^2 + x - 3. Substitute this into the given equation: (x^3 + 4x^2 + x - 3) + (3)/(2)x - x - 4 + 3 = 0 y + (1)/(2)x - 1 = 0 y = -(1)/(2)x + 1 Now, draw the line y = -(1)/(2)x + 1 on the same graph. To draw this line, find two points: If x = -4, y = -(1)/(2)(-4) + 1 = 2 + 1 = 3. Plot (-4, 3). If x = 0, y = -(1)/(2)(0) + 1 = 0 + 1 = 1. Plot (0, 1). If x = 2, y = -(1)/(2)(2) + 1 = -1 + 1 = 0. Plot (2, 0). The solutions to the equation x^3 + 4x^2 + (3)/(2)x - 4 = 0 are the x-coordinates of the intersection points of the curve y = x^3 + 4x^2 + x - 3 and the line y = -(1)/(2)x + 1. By observing the graph, the intersection points are approximately: x ≈ -3.8 x ≈ -1.2 x ≈ 1.0 Solutions: x ≈ -3.8, -1.2, 1.0 x ≈ -3.8, -1.2, 1.0