Manipulating RootsDate: 08/08/98 at 12:46:32 From: Vivian Wave Subject: Equations and Functions Hello: It would be great if you could help me to solve this problem. If a and b are the roots of the equation 3x^2 - 5x + 1 = 0, find the equation whose roots are a/b and b/a. I appreciate your assistance. Thank you. Date: 08/10/98 at 12:54:54 From: Doctor Peterson Subject: Re: Equations and Functions Hi, Vivian. This is an interesting question. You could, of course, just solve the original quadratic and then write an equation whose roots are a/b and b/a, but there's a nice trick that lets you do it more quickly. The basic idea is that if you have an equation of the form: A*x^2 + B*x + C = 0 and its two roots are a and b, then the equation can also be written as: A*(x - a)*(x - b) = 0 (where I've put in the factor A so the coefficient of x^2 will match the original equation), and this expands to: A*x^2 - A*(a+b)*x + A*a*b = 0 This means that: a+b = -B/A and a*b = C/A In your case, A = 3, B = -5, and C = 1, so we have: a+b = 5/3 and a*b = 1/3 Now see if you can reverse this and find what A, B, and C have to be (you can take A = 1) if the roots are a/b and b/a. (Hint: the product of the roots of this new equation will be a b - * - = 1 b a and the sum of the roots will be a b b + a - + - = ----- b a b * a and you can figure these out easily.) - Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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