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 find the values of a and b, we will use the given conditions to form a system of linear equations. Step 1: Use the condition g(2) = 4. Substitute x=2 into the function g(x) = ax^2 + bx - 3: g(2) = a(2)^2 + b(2) - 3 g(2) = 4a + 2b - 3 Since g(2) = 4, we set the expression equal to 4: 4a + 2b - 3 = 4 4a + 2b = 7 (Equation 1) Step 2: Use the condition g(-2) = 6. Substitute x=-2 into the function g(x) = ax^2 + bx - 3: g(-2) = a(-2)^2 + b(-2) - 3 g(-2) = 4a - 2b - 3 Since g(-2) = 6, we set the expression equal to 6: 4a - 2b - 3 = 6 4a - 2b = 9 (Equation 2) Step 3: Solve the system of linear equations. We have the system: 1) 4a + 2b = 7 2) 4a - 2b = 9 Add Equation 1 and Equation 2 to eliminate b: (4a + 2b) + (4a - 2b) = 7 + 9 8a = 16 a = (16)/(8) a = 2 Step 4: Substitute the value of a into Equation 1 to find b. Using Equation 1: 4a + 2b = 7 Substitute a=2: 4(2) + 2b = 7 8 + 2b = 7 2b = 7 - 8 2b = -1 b = -(1)/(2) The values are a=2 and b=-(1)/(2). That's 2 down. 3 left today — send the next one.