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
To solve the system of equations: x + y = 4 (1) xy = 16 (2) Step 1: Express y in terms of x from equation (1). From x + y = 4, we get: y = 4 - x Step 2: Substitute this expression for y into equation (2). x(4 - x) = 16 Step 3: Expand and rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 4x - x^2 = 16 x^2 - 4x + 16 = 0 Step 4: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. For x^2 - 4x + 16 = 0, we have a=1, b=-4, c=16. x = -(-4) ± sqrt((-4)^2 - 4(1)(16))2(1) x = 4 ± sqrt(16 - 64)2 x = 4 ± sqrt(-48)2 Step 5: Simplify the square root of the negative number. Recall that sqrt(-1) = i and sqrt(48) = sqrt(16 × 3) = 4sqrt(3). x = 4 ± 4sqrt(3)i2 x = 2 ± 2sqrt(3)i This gives two possible values for x: x_1 = 2 + 2sqrt(3)i x_2 = 2 - 2sqrt(3)i Step 6: Find the corresponding values for y using y = 4 - x. For x_1 = 2 + 2sqrt(3)i: y_1 = 4 - (2 + 2sqrt(3)i) y_1 = 4 - 2 - 2sqrt(3)i y_1 = 2 - 2sqrt(3)i For x_2 = 2 - 2sqrt(3)i: y_2 = 4 - (2 - 2sqrt(3)i) y_2 = 4 - 2 + 2sqrt(3)i y_2 = 2 + 2sqrt(3)i The solutions for the system of equations are: (x, y) = (2 + 2sqrt(3)i, 2 - 2sqrt(3)i) and (2 - 2sqrt(3)i, 2 + 2sqrt(3)i) That's 2 down. 3 left today — send the next one.