Solve equation by completing the square. Li

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Basic Funcs Trig

To solve the equation by completing the square, first, we may want to make the coefficient of equal to 1. Divide the entire equation by 3:

Now, isolate the constant on one side:

Next, complete the square. Take half of (which is ), square it (getting ), and add it to both sides:

Now you have:

Taking the square root of both sides gives you . Rearranging this yields:

Thus, the solutions are:

Did you know that completing the square isn’t just a mathematical trick? It has historical roots back to ancient civilizations like the Babylonians, who used geometric methods to solve quadratic equations. The process has evolved into our modern algebraic techniques and remains crucial in calculus, where we encounter integrals and solving optimization problems. Who knew history could be so pivotal in understanding today’s math!

Another fun fact about quadratics: many popular structures utilize parabolic shapes due to their nature of distributing weight. Think of bridges or even satellite dishes! By understanding how to manipulate these equations, architects and engineers can design more efficient and striking constructions. So next time you gaze up at a beautifully arched building, remember—there’s a quadratic equation at play behind its stunning form!