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: Define the numbers and set up the equations. Let the two numbers be x and y. The difference of the two numbers is 4: x - y = 4 (1) The product of the two numbers is 140: x · y = 140 (2) Step 2: Form a quadratic equation. From equation (1), express x in terms of y: x = y + 4 Substitute this expression for x into equation (2): (y+4)y = 140 Expand the equation: y^2 + 4y = 140 Rearrange the equation into the standard quadratic form ay^2 + by + c = 0: y^2 + 4y - 140 = 0 The quadratic equation is y^2 + 4y - 140 = 0. Step 3: Solve the quadratic equation for y. We can solve y^2 + 4y - 140 = 0 using the quadratic formula y = -b ± sqrt(b^2 - 4ac)2a. Here, a=1, b=4, c=-140. y = -4 ± sqrt(4^2 - 4(1)(-140))2(1) y = -4 ± sqrt(16 + 560)2 y = -4 ± sqrt(576)2 y = (-4 ± 24)/(2) This gives two possible values for y: y_1 = (-4 + 24)/(2) = (20)/(2) = 10 y_2 = (-4 - 24)/(2) = (-28)/(2) = -14 Step 4: Find the corresponding values for x. Using x = y + 4: If y = 10: x = 10 + 4 = 14 If y = -14: x = -14 + 4 = -10 Step 5: State the numbers. The two numbers are either 14 and 10 or -10 and -14. That's 2 down. 3 left today — send the next one.