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's the solution to Question 50, verifying the steps provided: Question 50: Given the equation: x^2 + px - 10 = 0 Assume the roots are 2 and -5. Find the value of p. For a quadratic equation in the form ax^2 + bx + c = 0, the following relationships hold for its roots r_1 and r_2: Sum of roots: r_1 + r_2 = -(b)/(a) Product of roots: r_1 × r_2 = (c)/(a) From the given equation x^2 + px - 10 = 0, we have: a = 1 b = p c = -10 The assumed roots are r_1 = 2 and r_2 = -5. Step 1: Check the product of the roots. The product of the given roots is: r_1 × r_2 = 2 × (-5) = -10 Using the formula for the product of roots from the equation: (c)/(a) = (-10)/(1) = -10 Since the calculated product matches the formula, the assumed roots are consistent with the constant term of the equation. (Note: The student's work incorrectly showed 2 + (-5) for the product, but the result -10 was correct for the product). Step 2: Calculate the sum of the roots. The sum of the given roots is: r_1 + r_2 = 2 + (-5) = -3 Step 3: Use the sum of roots formula to find p. We know that the sum of roots is also equal to -(b)/(a). r_1 + r_2 = -(b)/(a) Substitute the calculated sum and the values of a and b from the equation: -3 = -(p)/(1) -3 = -p Multiply both sides by -1: p = 3 Therefore, the value of p is 3. The final answer is p = 3.