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
Right, let's go. Step 1: Identify the given roots and the general form of the quadratic equation. The given quadratic equation is x^2 + bx + c = 0. The roots are x_1 = -7+sqrt(61)2 and x_2 = -7-sqrt(61)2. For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is x_1 + x_2 = -(b)/(a) and the product of the roots is x_1 · x_2 = (c)/(a). In this case, a=1. Step 2: Calculate the sum of the roots to find the value of b. x_1 + x_2 = -7+sqrt(61)2 + -7-sqrt(61)2 x_1 + x_2 = -7+sqrt(61) - 7 - sqrt(61)2 x_1 + x_2 = (-14)/(2) x_1 + x_2 = -7 Since x_1 + x_2 = -(b)/(a) and a=1: -7 = -b b = 7 Step 3: Calculate the product of the roots to find the value of c. x_1 · x_2 = (-7+sqrt(61)2) · (-7-sqrt(61)2) This is in the form (A+B)(A-B) = A^2 - B^2, where A = -7 and B = sqrt(61). x_1 · x_2 = (-7)^2 - (sqrt(61))^22 · 2 x_1 · x_2 = (49 - 61)/(4) x_1 · x_2 = (-12)/(4) x_1 · x_2 = -3 Since x_1 · x_2 = (c)/(a) and a=1: -3 = c The values are b=7 and c=-3. The final answer is b=7, c=-3. Send me the next one 📸