Ref: /abs/1910.06709 : A Simple Proof of the Quadratic FormulaĬorrection: We amended a sentence to say that the method has never been widely shared before and included a quote from Loh. Solving quadratics by completing the square. Worked example: completing the square (leading coefficient 1) Solving quadratics by completing the square: no solution. Either way, Babylonian tax calculators would surely have been impressed. Solve by completing the square: Non-integer solutions. To speed adoption, Loh has produced a video about the method. The question now is how widely it will spread and how quickly. The derivation emerged from this process. Suppose ax + bx + c 0 is the quadratic equation, then the formula to find the roots of this equation will be: x -b (b2-4ac)/2a. Since quadratics have a degree equal to two, therefore there will be two solutions for the equation. Loh, who is a mathematics educator and popularizer of some note, discovered his approach while analyzing mathematics curricula for schoolchildren, with the goal of developing new explanations. The formula for a quadratic equation is used to find the roots of the equation. “Perhaps the reason is because it is actually mathematically nontrivial to make the reverse implication: that always has two roots, and that those roots have sum −B and product C,” he says. So why now? Loh thinks it is related to the way the conventional approach proves that quadratic equations have two roots. None of them appear to have made this step, even though the algebra is simple and has been known for centuries. That's not the case with the other techniques The second coolest thing about the quadratic formula: it's easy to use. You can apply it to any quadratic equation out there and you'll get an answer every time. He has looked at methods developed by the ancient Babylonians, Chinese, Greeks, Indians, and Arabs as well as modern mathematicians from the Renaissance until today. The coolest thing about the formula is that it always works. Loh has searched the history of mathematics for an approach that resembles his, without success. Yet this technique is certainly not widely taught or known." If students can remember some simple generalizations about roots, they can decide where to go next.Loh says he "would actually be very surprised if this approach has entirely eluded human discovery until the present day, given the 4,000 years of history on this topic, and the billions of people who have encountered the formula and its proof. Loh believes students can learn this method more intuitively, partly because there’s not a special, separate formula required. It’s quicker than the classic foiling method used in the quadratic formula-and there’s no guessing required. When solving for u, you’ll see that positive and negative 2 each work, and when you substitute those integers back into the equations 4–u and 4+u, you get two solutions, 2 and 6, which solve the original polynomial equation. When you multiply, the middle terms cancel out and you come up with the equation 16–u2 = 12. For equations with real solutions, you can use the graphing tool to visualize the solutions. The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. So the numbers can be represented as 4–u and 4+u. Step 1: Enter the equation you want to solve using the quadratic formula. Suppose ax² + bx + c 0 is the quadratic equation, then the formula to find the roots of this equation will be: x -b± (b2-4ac)/2a. If the two numbers we’re looking for, added together, equal 8, then they must be equidistant from their average. The formula for a quadratic equation is used to find the roots of the equation. Instead of starting by factoring the product, 12, Loh starts with the sum, 8. Those two numbers are the solution to the quadratic, but it takes students a lot of time to solve for them, as they’re often using a guess-and-check approach. “Normally, when we do a factoring problem, we are trying to find two numbers that multiply to 12 and add to 8,” Dr.
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