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 August 13th, 2017, 04:29 AM #1 Newbie   Joined: Aug 2014 From: England Posts: 14 Thanks: 0 Math Focus: Calculus System of equations (help) Hi, I have a question I was hoping somebody could help with. I have two equations as follows: 2U + V = 7 UV =6 Now I could easily do this in my head and get the answer, but I want to solve it using the appropriate mathematical principles. If I turned the equation into a quadratic I could solve using a calculator and I get U = 1.5 and V = 4, or U =2 and V =3 However what if I did not use a calculator? How would I solve it then? What I did was rearrange the first equation to get V = 7 - 2U Plug that into the second equation to get U(7 - 2U) = 6 ... and expand to get -2U^2 + 7U - 6 = 0 And I cannot go any further in solving for U without using a calculator
 August 13th, 2017, 04:32 AM #2 Newbie   Joined: Aug 2014 From: England Posts: 14 Thanks: 0 Math Focus: Calculus @denis
 August 13th, 2017, 04:41 AM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out. $-2U^2 + 7U - 6 = 0$ Factor U $U(7 - 2U) - 6 = 0$ What values of U will cause the above to be true? Alternatively, you could use the quadratic formula. Have you learnt about that? Last edited by Joppy; August 13th, 2017 at 04:43 AM.
 August 13th, 2017, 04:41 AM #4 Math Team     Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 878 Thanks: 60 Math Focus: सामान्य गणित Use quadratic formula $\displaystyle U = \frac {-b\pm \sqrt {b^{2} -4ac}}{2a}$ For $\displaystyle aU^{2} +bU + c =0$
August 13th, 2017, 05:31 AM   #5
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Quote:
 Originally Posted by srahman33 @denis
WHAT did I do now?

August 13th, 2017, 06:52 AM   #6
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Math Focus: Yet to find out.
Quote:
 Originally Posted by Denis WHAT did I do now?
Tsk tsk!

 August 13th, 2017, 07:34 AM #7 Global Moderator   Joined: Dec 2006 Posts: 18,965 Thanks: 1606 For distinct numbers p and q, the expression (x - p)(x - q) has two zeros: p and q. The expression can be written as x² - (p + q)x + pq. For the posted problem, p = 2U and q = V, so p + q = 7 and pq = 2UV = 12. Can one convert the expression x² - 7x + 12 into the form (x - p)(x - q)? This would be a bit awkward if p and q can't be integers. If, however, they can be integers, their product is 12, and so there are only a very few possibilities to try. Those that work are (p, q) = (3, 4) and (p, q) = (4, 3), because the sum of p and q is then 7. As U = p/2 and V = q, the required solutions are (U, V) = (3/2, 4) and (U, V) = (2, 3).

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