What is the value of k ?

Find the value of color(red)k for which the inequality color(red)(x^2-2(4k-1)x+15k^2-2k-7>0 is valid for ANY value of color(red)x

1 Answer
Jun 16, 2018

The answer is k in (2,4)

Explanation:

For a quadratic equation f(x)>0, the graph must touch the x-axis

The discriminant must be =0

f(x)=x^2-2(4k-1)x+15k^2-2k-7

Delta=(2(4k-1))^2-4(1)(15k^2-2k-7)=0

=>, 4(16k^2-8k+1)-60k^2+8k+28=0

=>, 64k^2-32k+4-60k^2+8k+28=0

=>, 4k^2-24k+32=0

=>, 4(k^2-6k+8)=0

=>, (k-2)(k-4)=0

=>, {(k=2),(k=4):}

When k=2

=>, x^2-14x+49=0

graph{x^2-14x+49 [-14.48, 21.55, -4.62, 13.4]}

When k=4

=>, x^2-30x+225=0

graph{x^2-30x+225 [-14.48, 21.55, -4.62, 13.4]}

When k=3

=>, x^2-22x+122=0

graph{x^2-22x+122 [-14.48, 21.55, -4.62, 13.4]}