Solve. \[ b^{2}-10 b+25=0 \]

This equation can be solved by factoring. Notice that the left side can be rewritten as a perfect square: \[ (b - 5)^2 = 0 \] To find the value of \(b\), we take the square root of both sides: \[ b - 5 = 0 \] Solving for \(b\) gives us: \[ b = 5 \] So the solution to the equation \(b^{2}-10 b+25=0\) is \(b = 5\). Now, let's explore a bit of history about quadratic equations! They date back to ancient civilizations like the Babylonians who had methods to solve certain quadratic equations around 2000 BC. They mainly focused on geometric interpretations rather than algebraic solutions, much like the way you might visualize solving a puzzle! And if you’re interested in real-world applications, quadratic equations show up everywhere! From projectile motion, like calculating the path of a thrown ball, to finance when evaluating profit maximization problems. Understanding how to solve them can provide key insights into everyday scenarios!