Working with Quadratic RootsDate: 08/15/2002 at 10:22:53 From: Mandeep Singh Subject: Quadratic equations Hi, Could you help me with this question: Given that a and b are the roots of the quadratic equation x^ - 5x + 3 = 0 form, without solving the equation, the quadratic equation whose roots are a b - - b a Please could you show me step by step how to do this? Date: 08/15/2002 at 12:24:35 From: Doctor Peterson Subject: Re: Quadratic equations Hi, Mandeep. Since a and b are the roots, you know that x^2 - 5x + 3 = (x - a)(x - b) An equation with roots a/b and b/a will be (x - a/b)(x - b/a) = 0 Try expanding this into a quadratic in standard form, and compare with the original, using what you know about the sum and product of its roots. If you need more help, please write back and show me what you found. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/16/2002 at 18:15:08 From: Doctor Greenie Subject: Re: Quadratic equations Hi, Mandeep - I find this kind of problem to be kind of fun, once you have done a couple and see the general procedure. If a and b are the roots of a quadratic equation, then the equation is (x-a)(x-b) = 0 So with the given equation x^2-5x+3 = 0 we have (x-a)(x-b) = x^2 - (a+b)x + ab = x^2 - 5x + 3 = 0 Equating coefficients of the linear terms, we see that -(a+b) = -5 (or a+b = -(-5) = 5) And equating the constant terms, we see that ab = 3 What we have shown is that if we have a quadratic equation with leading coefficient 1 (i.e., the coefficient of the x^2 term is 1), then the sum of the two roots is the opposite of the coefficient of the linear term, and the product of the two roots is the constant term. This is equivalent to properties with which you may already be familiar: Given a quadratic equation px^2+qx+r = 0 (1) the product of the roots is r/p (2) the sum of the roots is -q/p In your problem, we are supposed to find a quadratic equation whose roots are a/b and b/a. We know this quadratic equation will be of the form x^2 + Ax + B = 0 where the coefficient A is the opposite of the sum of the two roots and the constant term B is the product of the two roots. The sum of the two given roots (a/b and b/a) is (a^2+b^2)/ab; the product of these two roots is 1. Thus we know that the equation we are looking for will be x^2 - ((a^2+b^2)/ab)x + 1 = 0 We will be done with the problem if we can evaluate the expression (a^2+b^2)/ab We can do this because a^2+b^2 a^2+2ab+b^2 - 2ab (a+b)^2 - 2ab 5^2 - 2(3) 19 ------- = ----------------- = ------------- = ---------- = -- ab ab ab 3 3 So the equation we are looking for is x^2 - (19/3)x + 1 = 0 A little messy algebra - solving the original equation to find the roots a and b, then calculating a/b and b/a, then forming the equation with roots a/b and b/a - will show this to be the correct result. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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