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Find the equation of a tangent line of a graph pf y=x^2+4lnx, which is parallel to y=6x-3?

I know parallel lines have the same slope so the slope is 6 but I do not know how to find the b value.

  • Calc -

    Well you need to find where on that nasty function the slope is 6. Then at that (x,y) you write the y = 6 x + b that goes through that point.

  • Calc -

    how do I do this?

  • Calc -

    y = x^2 + 4 ln x
    dy/dx = slope = 2 x +4/x
    when is that slope 6?
    6 = 2 x + 4/x
    I get at x = 1 and at x = 2
    well 1 is easy because ln 1 = 0. It just said "a" tangent line, not all tangent lines.
    then y = 1
    so I am going to use (1,1)

  • Calc -

    I understand what you are saying but how do you do the algebra foe 6=2x+4/x I seem to have forgotten my algebra.

  • Calc -

    actually it's not so bad

    the derivative of your function is
    2x + 4/x
    setting this equal to 6 gave me a quadratic which factored nicely and had roots of
    x = 1 or x=2
    if x=1 then y = 1+4ln1
    y=1 ,
    so one case is m=6, point (1,1)

    case 2
    x=2, then y = 4 + 4ln2

    so m=6, point (2,4+4ln2)


  • Calc -

    All set now?

  • Calc -

    No i need help understanding the algebra behind solving 2x+4/x=6

  • Calc -

    multiply both sides by x
    6 x = 2 x^2 + 4
    2 x^2 - 6 x + 4 = 0
    1 x^2 - 3 x + 2 = 0
    (x-1)(x-2) = 0
    x = 1
    x = 2 are solutions.
    I used 1 so the point I could take was (1,1)

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