What are the value of $k$ , for which $x^2+5kx+16=0$ has no real roots?

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Answer 1 · 2016-11-02T07:08:06

The values of k for which the equation $x^2+5kx+16=0$ has no real roots are $-8/5 < k < 8/5$

An equation $ax^2+bx+c=0$ has no real roots if the determinant $b^2-4ac<0$

$x^2+5kx+16=0$ has no real roots if the determinant $(5k)^2-4xx1xx16=25k^2-64<0$ or

$(5k-8)(5k+8)<0$

As $(5k-8)(5k+8)$ is negative,

either $5k+8<0$ and $5k-8>0$

But this is just not possible, as $k$ cannot be $k<-8/5$ as well as $k>8/5$

Other alternative is $5k+8>0$ and $5k-8<0$ i.e.

$k> -8/5$ as well as $k<8/5$ . This is possible if value of $k$ is between $-8/5$ and $8/5$ .

Hence $-8/5 < k < 8/5$

What are the value of kk, for which x^2+5kx+16=0x^2+5kx+16=0 has no real roots?