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Solution to quadratic equation

  1. Jun 30, 2016 #1
    Q48_49hall_XXVIC.jpg 1. The problem statement, all variables and given/known data


    2. Relevant equations


    3. The attempt at a solution

    I just want some clarification about the rules on solving quadratic equation. My question is coming from my solution for problem 49.

    Solving problem 49

    Factor: (x−p−q)(√(x−q)+√(x−p))=0
    Therefore: x−p−q=0⇒x=p+q this agrees with the answer in my book

    What Im confused about is the part where I divided out the lhs by (√(x−q)+√(x−p)). This factor contains x in it and I'm worried that I may have discarded the other root for the equation. And since it is a quadratic equation I am expecting to get two roots for the equation. Can you enlighten me about this matter? Thanks.
     
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  3. Jun 30, 2016 #2

    ehild

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    When you solve an equation of form f(x) g(x) =0, you use the fact that the product is zero if either f(x)=0 or g(x)=0. In principle, it is possible that √(x−q)+√(x−p) = 0. What does it mean for √(x−q) and √(x−p)?
    You started the solution by multiplying the whole equation with √(x−q)√(x−p). What is the condition that you may do it?
     
  4. Jun 30, 2016 #3
    The one and only posibility that √(x−q)+√(x−p) =0 is if x=p=q. Therefore x=q=p is also a solution? Im still confused. Please bear with me.
     
  5. Jun 30, 2016 #4

    Ray Vickson

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    You are not allowed to have either ##x = p## or ##x = q##, because the right-hand-side of your original equation would then contain division by zero, which is absolutely forbidden.
     
  6. Jun 30, 2016 #5
    What? When x = x+p we will also have division by zero right? Why is it that √(x-q)+√(x-p) is not allowed to be equal zero but x-p-q can be equal zero?
     
  7. Jun 30, 2016 #6

    SammyS

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    The following doesn't make sense unless p = 0 .
     
  8. Jun 30, 2016 #7

    Charles Link

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    Suggestion: Group the sqrt (x-p ) terms on one side of the equation (one term will be in the denominator), and the sqrt(x-q) terms on the other side of the equation and square both sides of the equation. You can then do some additional algebra. It's then a couple more steps, but you do get a quadratic expression for x in terms of p and q. The quadratic expression does factor and simplify and it does have a simple solution.
     
    Last edited: Jun 30, 2016
  9. Jul 1, 2016 #8

    ehild

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    You started the solution with multiplying the whole equation by √(x−q)*√(x−p). Before multiplying an equation with an expression you have to exclude that it is zero.
     
  10. Jul 1, 2016 #9

    Ray Vickson

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    Please read responses carefully: I said ".... original equation...". In your original equation you had
    [tex] \text{original right-hand-side} = \frac{q}{\sqrt{x-p}} + \frac{p}{\sqrt{x-q}} [/tex]
    If you take ##x = p## or ##x = q## you will have ##1/\sqrt{0}## on the right, and that is forbidden.

    Of course, after you multiply your equation by ##\sqrt{x-p} \sqrt{x-q}## you would have
    [tex] \text{new right-hand-side} =q \sqrt{x-q} + p \sqrt{x-p} [/tex]
    and in that new rhs you are certainly allowed to have ##x = p## or ##x = q##. However, that would not be giving you a solution to the original equation.
     
    Last edited: Jul 1, 2016
  11. Jul 1, 2016 #10
    Thank you now it is clear!
     
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