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 express y = 2x^2 + 11x + 6 in the form a(x+b)^2 + c and find its minimum value, we use the method of completing the square. Step 1: Factor out the coefficient of x^2 from the terms involving x. y = 2(x^2 + (11)/(2)x) + 6 Step 2: Complete the square inside the parenthesis. To do this, take half of the coefficient of x ((11)/(2)), square it, and add and subtract it within the parenthesis. Half of (11)/(2) is (11)/(4). Squaring (11)/(4) gives ((11)/(4))^2 = (121)/(16). y = 2(x^2 + (11)/(2)x + (121)/(16) - (121)/(16)) + 6 Step 3: Group the perfect square trinomial and distribute the 2. y = 2((x + (11)/(4))^2 - (121)/(16)) + 6 y = 2(x + (11)/(4))^2 - 2((121)/(16)) + 6 y = 2(x + (11)/(4))^2 - (121)/(8) + 6 Step 4: Combine the constant terms. y = 2(x + (11)/(4))^2 - (121)/(8) + (48)/(8) y = 2(x + (11)/(4))^2 - (73)/(8) This is in the form a(x+b)^2 + c, where a=2, b=(11)/(4), and c=-(73)/(8). Step 5: Find the minimum value. Since a=2 is positive, the parabola opens upwards, and the vertex represents the minimum point. The minimum value of the expression a(x+b)^2 + c occurs when (x+b)^2 = 0, which means x = -b. At this point, the value of y is c. The minimum value is -(73)/(8). The expression in the form a(x+b)^2 + c is: y = 2(x + (11)/(4))^2 - (73)/(8) The minimum value is: -(73)/(8)