Find the set of values of k for which the line y=kx-4 intersects the curve y=x²-2x at 2 distinct points?

Question

Find the set of values of k for which the line y=kx-4 intersects the curve y=x²-2x at 2 distinct points?

in progress 0
Ngọc Hoa 4 years 2021-08-25T02:23:09+00:00 1 Answers 550 views 0

Answers ( )

  1. vankhanh
    0
    2021-08-25T02:24:42+00:00
    Reply

    Answer:

    -6 < k < 2

    Step-by-step explanation:

    Given

    y = x^2 - 2x

    y =kx -4

    Required

    Possible values of k

    The general quadratic equation is:

    ax^2 + bx + c = 0

    Subtract y = x^2 - 2x and y =kx -4

    y - y = x^2 - 2x - kx +4

    0 = x^2 - 2x - kx +4

    Factorize:

    0 = x^2 +x(-2 - k) +4

    Rewrite as:

    x^2 +x(-2 - k) +4=0

    Compare the above equation to: ax^2 + bx + c = 0

    a = 1

    b= -2-k

    c =4

    For the equation to have two distinct solution, the following must be true:

    b^2 - 4ac > 0

    So, we have:

    (-2-k)^2 -4*1*4>0

    (-2-k)^2 -16>0

    Expand

    4 +4k+k^2-16>0

    Rewrite as:

    k^2 + 4k - 16 + 4 >0

    k^2 + 4k - 12 >0

    Expand

    k^2 + 6k-2k - 12 >0

    Factorize

    k(k + 6)-2(k + 6) >0

    Factor out k + 6

    (k -2)(k + 6) >0

    Split:

    k -2 > 0 or k + 6> 0

    So:

    k > 2 or k > -6

    To make the above inequality true, we set:

    k < 2 or k >-6

    So, the set of values of k is:

    -6 < k < 2

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )