We already have $ a - b = 6 $ and $ a^2 + b^2 = 130 $. We need $ ab $.
["Solving for $ ab $ Given $ a - b = 6 $ and $ a^2 + b^2 = 130 $", "Understanding relationships between variables is key in algebra, especially when working with equations involving sums and products. If you're given $ a - b = 6 $ and $ a^2 + b^2 = 130 $, finding the product $ ab $ becomes straightforward by using a powerful identity in algebra.", "---", "### What’s the identity we can use?", "We know the following algebraic identity:\n$$\n(a - b)^2 = a^2 - 2ab + b^2\n$$", "This identity connects the difference of $ a $ and $ b $, their squares, and their product $ ab $.", "---", "### Step 1: Use the given value of $ a - b $", "Given:\n$$\na - b = 6\n$$\nSquare both sides:\n$$\n(a - b)^2 = 6^2 = 36\n$$", "So:\n$$\na^2 - 2ab + b^2 = 36\n$$", "---", "### Step 2: Substitute the known value of $ a^2 + b^2 $", "We are told:\n$$\na^2 + b^2 = 130\n$$", "Substitute this into the equation from Step 1:\n$$\n(a^2 + b^2) - 2ab = 36\n\Rightarrow 130 - 2ab = 36\n$$", "---", "### Step 3: Solve for $ ab $", "Subtract 130 from both sides:\n$$\n-2ab = 36 - 130 = -94\n$$", "Divide by $-2$:\n$$\nab = \frac{-94}{-2} = 47\n$$", "---", "### Final Answer:\n$$\nab = 47\n$$", "---", "### Why this matters\nKnowing $ ab $ is essential when solving quadratic equations or working with symmetric expressions. In real-world applications—like physics, engineering, or data modeling—this product helps describe relationships between variables efficiently.", "So, whenever you have $ a - b $ and $ a^2 + b^2 $, just use the identity $ (a - b)^2 = a^2 - 2ab + b^2 $ to find $ ab $ quickly and accurately."]
