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: ax^{2} + 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 (b^{2})/(4a^{2}) 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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