The Quadratic FormulaDate: 12/12/2005 at 00:04:25 From: Chris Subject: The Quadratic formula We've been studying the quadratic formula as a way to solve quadratic equations, but I'm a little confused by it. Can you give me a summary of what the formula is, how and when you apply it, and why it's important? Date: 12/12/2005 at 11:36:11 From: Doctor Ian Subject: Re: The Quadratic formula Hi Chris, _When_ you apply it is: When it's difficult or impossible to factor the quadratic expression in the equation you are trying to solve. Usually, this means when the coefficient of x^2 is something other than 1, or when the roots aren't integers. _How_ you apply it is, you write down the formula, -b +/- sqrt(b^2 - 4ac) x = ---------------------- when ax^2 + bx + c = 0 2a and then you figure out what a, b, and c are. For example: 0 = 3x^2 - 4x - 9 0 = 3x^2 + -4x + -9 | | |____ c = -9 | |_________ b = -4 |_________________ a = 3 Then you substitute into the formula, using ()'s around the values to keep from messing up any signs: -(-4) +/- sqrt((-4)^2 - 4(3)(-9)) x = --------------------------------- 2(3) and then you simplify that. You will usually get two answers, although sometimes you can get one answer twice. You can tell how many and what type of answers you will get by the expression in the square root, which is called the "discriminant". If the "discriminant" evaluates to zero, you just get one root: -b x = -- 2a If it evaluates to a positive perfect square, you get a couple of rational roots: -b + n -b - n x = ------, ------ 2a 2a If it evaluates to a positive non-square, you get a couple of irrational roots: _ _ -b + \/n -b - \/n x = --------, -------- 2a 2a And if it evaluates to a negative number, you get a couple of imaginary roots: __ __ -b + \/-n -b - \/-n x = ---------, --------- 2a 2a _ _ -b + i*\/n -b - i*\/n = ----------, ---------- where i^2 = -1 2a 2a (Can you see why the name "discriminant" was chosen?) Note that sometimes you can _partially_ apply it when you're trying to factor, just to see if you can expect factoring to work. For example, if I'm looking at something like 0 = 3x^2 + -4x + -9 | | |____ c = -9 | |_________ b = -4 |_________________ a = 3 I'll quickly evaluate the discriminant, b^2 - 4ac = (-4)^2 - 4(3)(-9) = 16 + 128 = 124 Since this isn't a perfect square, I know I'm not going to get factoring to work out, so I'll go ahead and apply the rest of the formula. On the other hand, if I'm looking at something like 2x^2 + -4x + -16 the discriminant is b^2 - 4ac = (-4)^2 - 4(2)(-16) = 16 + 128 = 144 = 12^2 so that tells me that factoring will work. Of course, at this point, it's easier to apply the rest of the formula to get the roots, -(-4) +/- 12 ------------ = 1 +/- 3 = -2 or 4 4 so depending on what I'm trying to do, it might not be worth going back to complete the factoring. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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