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Proof of the quadratic formulaThe following is a proof of the quadratic formula. It will show you how the quadratic formula that is widely used was developed. The proof is done using the standard form of a quadratic equation and solving the standard form by completing the square Start with the the standard form of a quadratic equation: ax2 + bx + c = 0 Divide both sides of the equation by a so you can complete the square   Subtract c/a from both sides   Complete the square: The coefficient of the second term is b/a Divide this coefficient by 2 and square the result to get (b/2a)2 Add (b/2a)2 to both sides:  Since the left side of the equation right above is a perfect square, you can factor the left side by using the coefficient of the first term (x) and the base of the last term(b/2a) Add these two and raise everything to the second. Then, square the right side to get (b2)/(4a2)  Get the same denominator on the right side:   Now, take the square root of each side:  Simplify the left side:   Rewrite the right side:  Subtract b/2a from both sides:   Adding the numerator and keeping the same denominator, we get the quadratic formula:  The + - between the b and the square root sign means plus or negative. In other words, most of the time, you will get two answers when using the quadratic formula.
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