5.1.1 \( \quad x^{2}-2 x-35=0 \)

To solve the equation \( x^{2} - 2x - 35 = 0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a = 1, b = -2, c = -35 \). Plugging in these values, first calculate the discriminant: \( b^{2} - 4ac = (-2)^{2} - 4(1)(-35) = 4 + 140 = 144 \). Now, substituting back into the formula yields \( x = \frac{2 \pm \sqrt{144}}{2(1)} = \frac{2 \pm 12}{2} \). This gives us two solutions: \( x = 7 \) and \( x = -5 \). Solving quadratic equations has been an important skill throughout history, particularly since ancient civilizations used similar methods! The Babylonians, for example, had geometric methods for finding roots of equations as far back as 2000 BC. Fast forward, and here we find ourselves using algebraic methods and formulas developed over centuries to unravel the mysteries of numbers. In real-world applications, solving quadratic equations can help in various fields like physics, finance, and engineering. For instance, it's essential for calculating projectile motion in sports, determining the optimal dimensions for a product, or even in calculating profits where revenue and costs can be modeled by quadratic functions. If you're ever planning to build a water fountain or design a parabolic bridge, understanding how to handle those curves will come in handy!