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: Find the values of a and b for the first equation. The given quadratic equation is y = x^2 + ax + b. The axis of symmetry for a quadratic y = Ax^2 + Bx + C is given by the formula x = -(B)/(2A). In this equation, A=1 and B=a. So, the axis of symmetry is x = -(a)/(2). We are given that the axis of symmetry is x = -2. -(a)/(2) = -2 a = 4 The graph passes through the point (-1, -5). Substitute x=-1, y=-5, and a=4 into the equation y = x^2 + ax + b: -5 = (-1)^2 + 4(-1) + b -5 = 1 - 4 + b -5 = -3 + b b = -5 + 3 b = -2 The values are a=4 and b=-2. Step 2: Find the value of k for the second equation. The given equation is y = (x+k)^2 + 3x - 2y. First, rearrange the equation into the standard quadratic form y = Ax^2 + Bx + C: y = x^2 + 2kx + k^2 + 3x - 2y Add 2y to both sides: 3y = x^2 + (2k+3)x + k^2 Divide by 3: y = (1)/(3)x^2 + (2k+3)/(3)x + (k^2)/(3) For a quadratic equation y = Ax^2 + Bx + C, the minimum value (since A = (1)/(3) > 0) occurs at the vertex and is given by C - (B^2)/(4A). In this equation, A = (1)/(3), B = (2k+3)/(3), and C = (k^2)/(3). The minimum value is: (k^2)/(3) - ((2k+3)/(3))^24((1)/(3)) = (k^2)/(3) - ((2k+3)^2)/(9)(4)/(3) = (k^2)/(3) - ((2k+3)^2)/(9) · (3)/(4) = (k^2)/(3) - ((2k+3)^2)/(12) We are given that the minimum value is -4. (k^2)/(3) - ((2k+3)^2)/(12) = -4 Multiply the entire equation by 12 to eliminate the denominators: 4k^2 - (2k+3)^2 = -48 Expand (2k+3)^2: 4k^2 - (4k^2 + 12k + 9) = -48 4k^2 - 4k^2 - 12k - 9 = -48 -12k - 9 = -48 Add 9 to both sides: -12k = -48 + 9 -12k = -39 Divide by -12: k = (-39)/(-12) k = (13)/(4) The final answers are a=4, b=-2 and k=(13)/(4).