Solve for \( a \) and \( b \) \( a-b=5 \) and \( a b=6 \)

To solve for \( a \) and \( b \) based on the equations \( a - b = 5 \) and \( ab = 6 \), we can start by expressing \( a \) in terms of \( b \) using the first equation: 1. From \( a - b = 5 \), we can express \( a \) as: \[ a = b + 5 \] 2. Now, substitute \( a \) in the second equation \( ab = 6 \): \[ (b + 5)b = 6 \] 3. Expanding this, we get: \[ b^2 + 5b - 6 = 0 \] 4. Now we need to factor or use the quadratic formula to solve this quadratic equation. We look for factors of \(-6\) that add up to \(5\). The equation can be factored as: \[ (b + 6)(b - 1) = 0 \] 5. Setting each factor to zero gives us: \[ b + 6 = 0 \quad \Rightarrow \quad b = -6 \] \[ b - 1 = 0 \quad \Rightarrow \quad b = 1 \] 6. Now, we can find the corresponding values of \( a \): - If \( b = -6 \): \[ a = -6 + 5 = -1 \] - If \( b = 1 \): \[ a = 1 + 5 = 6 \] So, the solutions are: 1. \( (a, b) = (-1, -6) \) 2. \( (a, b) = (6, 1) \)