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Malhar Rajpal The quadratic formula is used by most secondary students to solve quadratic equations in the form ax^2 + bx + c = 0. But most of these students don’t know where the formula comes from. In this article, I will explain the derivation of the quadratic formula. The idea of the quadratic formula is to isolate x from the initial equation, ax^2 + bx + c = 0. We can subtract c from both sides of the equation to get ax^2 + bx = -c. We then proceed by dividing both sides by the coefficient of x^2, namely a: The next step requires some creativity since these steps are unintuitive and rather inventive. Firstly, we must identify the coefficient of the linear (x) term in the above equation. This is b/a. We then divide this term by 2 and square it to get: (b/2a)^2=(b^2)/(4a^2). We add this term to both sides of the above equation to get: That was the creative step. We can now simplify the equation by making the terms on the RHS of the same denominator, and by completing the square on the LHS: We now take the square root of both sides: Finally we subtract b/2a from both sides to get the quadratic formula: Now you know how to derive the quadratic formula, you can use it in a classroom situation without worrying whether it actually makes sense!
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