Math

Question Find the value(s) of kk for which the quadratic equation y=kx24x+ky=k x^{2}-4 x+k has no real roots.

Studdy Solution

STEP 1

Assumptions1. The relation is given as y=kx4x+ky=k x^{}-4 x+k . We are looking for the value(s) of kk such that the relation has no zeroes. This means that the quadratic equation kx4x+k=0k x^{}-4 x+k=0 has no real roots.

STEP 2

A quadratic equation of the form ax2+bx+c=0ax^{2}+bx+c=0 has no real roots if its discriminant b24acb^{2}-4ac is less than zero.

STEP 3

In our given equation, a=ka=k, b=b=-, and c=kc=k. So, we can write the discriminant as=()2(k)(k)=(-)^{2}-(k)(k)

STEP 4

implify the discriminant.
=164k2=16-4k^{2}

STEP 5

For the quadratic equation to have no real roots, the discriminant must be less than zero.
164k2<016-4k^{2}<0

STEP 6

olve the inequality for kk. First, divide all terms by4.
4k2<04-k^{2}<0

STEP 7

Rearrange the inequality.
k24>0k^{2}-4>0

STEP 8

This inequality represents a quadratic equation that opens upwards and has roots at k=2k=-2 and k=2k=2. The inequality is satisfied for values of kk that fall outside of this interval.
k<2ork>2k<-2 \quad or \quad k>2So, the relation y=kx24x+ky=k x^{2}-4 x+k will have no zeroes for k<2k<-2 or k>2k>2.

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