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
Here are the solutions to the problems: 1.1.3 To solve the inequality x^2 + 2x - 24 0, we first find the roots of the corresponding quadratic equation. Step 1: Find the roots of the quadratic equation x^2 + 2x - 24 = 0. We can factor the quadratic expression: (x+6)(x-4) = 0 The roots are x = -6 and x = 4. Step 2: Determine the intervals where the inequality holds. Since the parabola y = x^2 + 2x - 24 opens upwards (the coefficient of x^2 is positive), the expression is greater than or equal to zero outside the roots. Therefore, the solution is when x is less than or equal to the smaller root or greater than or equal to the larger root. x -6 or x 4 The solution is x -6 or x 4. 1.2 To solve the system of equations simultaneously for x and y: 1) 5x + y = 3 2) 3x^2 - 2xy = y^2 - 105 Step 1: Express y in terms of x from the linear equation (1). y = 3 - 5x Step 2: Substitute this expression for y into the quadratic equation (2). 3x^2 - 2x(3 - 5x) = (3 - 5x)^2 - 105 Step 3: Expand and simplify the equation. 3x^2 - 6x + 10x^2 = (9 - 30x + 25x^2) - 105 13x^2 - 6x = 25x^2 - 30x - 96 Move all terms to one side to form a standard quadratic equation: 0 = 25x^2 - 13x^2 - 30x + 6x - 96 0 = 12x^2 - 24x - 96 Divide the entire equation by 12 to simplify: x^2 - 2x - 8 = 0 Step 4: Solve the quadratic equation for x. Factor the quadratic expression: (x-4)(x+2) = 0 This gives two possible values for x: x = 4 or x = -2 Step 5: Substitute these x values back into the linear equation y = 3 - 5x to find the corresponding y values. For x = 4: y = 3 - 5(4) = 3 - 20 = -17 For x = -2: y = 3 - 5(-2) = 3 + 10 = 13 The solutions are the pairs (x, y): (4, -17) and (-2, 13) Send me the next one 📸