Having trouble with questions on quadratic inequalities

Question

I am currently having trouble with these two questions on quadratic inequalities.

What is the set of values of $p$ for which $p(x^2+2) < 2x^2+6x+1$ for all real values

I tried converting the inequality to:

$p < \frac{2x^2+6x+1}{x^2+2}$

And then I tried to simplify it. However, I couldn't

The function f is defined by $f:x\mapsto x^2+kx+9$ for $x \in R$ Find the range of values of $k$ for which the range of $f$ is $f(x) \ge 0$

So I realised that $x^2+kx+9 \ge 0$ and then I'm stuck.

Can anyone help me with these two questions? Thanks in advance.

Answer 1

Let $k=\dfrac{2x^2+6x+1}{x^2+2}$

$\iff x^2(2-k)+6x+1-2k=0$

As $x$ is real, the discriminant of the above quadratic equation must be $\ge0$

$$\implies(2-k)^2-4(2-k)(1-2k)\ge0$$

$$\iff-7k^2+16k-4\ge0$$

$$\iff0\ge7k^2-16k+4=(7k-2)(k-2)$$

min$\left(\dfrac27,2\right)\le k\le$max$\left(\dfrac27,2\right)$

Answer 2

The condition that a quadratic is negative for all real $x$ implies that its graph never intersects the real axis, which implies it has no real roots. The quadratic $ax^2+bx+c$ has no real root if $b^2-4ac<0$.

1. $ax^2+bx+c$ will be $<0$ for all real $x$ is $b^2-4ac<0$ and $a<0$
2. $ax^2+bx+c$ will be $>0$ for all real $x$ is $b^2-4ac<0$ and $a>0$