1. ## discriminant with quadratic inequalities

Need help with this questions as soon as possible please.

1. find the range of values that b can take if x^2 + 5bx + b is positive for all real x.
2. use the discriminant to find the range of values k can take for kx^5 + 3x + k -4 = 0 to have two real distinct roots.

2. Hello, Tom G!

Find the range of values that $b$ can take
if $x^2 + 5bx + b$ is positive for all real $x$.

The function $y \:=\:x^2 + 5bx + b$ is an up-opening parabola.

Its vertex is at: . $x \:=\:\frac{-5b}{2}$
. . and: $y \:=\:\left(-\frac{5b}{2}\right)^2 + 5b\left(-\frac{5b}{2}\right) + b\:=\:\frac{b(4-25b)}{4}$

Hence, its vertex (lowest point) is: . $\left(-\frac{5b}{2},\:\frac{b(4-25b)}{4}\right)$

If the parabola is to be above the x-axis, then: . $\frac{b(4-25b)}{4} \:>\:0$

There are two cases:

$[1]\;\;b < 0$ and $4 - 25b \,< \,0$
. . .But this gives us: . $b < 0$ and $b > \frac{4}{25}$ . . . impossible

$[2]\;\;b > 0$ and $4 - 25b \,> \,0$
. . .which gives us: . $b > 0$ and $b < \frac{4}{25}$

Therefore, the range is: . $\boxed{0 \,< \,b \,< \,\frac{4}{25}}$

3. Originally Posted by Tom G
Need help with this questions as soon as possible please.

1. find the range of values that b can take if x^2 + 5bx + b is positive for all real x.
2. use the discriminant to find the range of values k can take for kx^5 + 3x + k -4 = 0 to have two real distinct roots.
2.

For the quadratic to have two distinct real roots the discriminant $b^2 - 4ac > 0$

(the other two possibilities are equal roots $b^2 - 4ac = 0$
or no real roots $b^2 - 4ac < 0$

So $a = k$, $b = 3$ and $c = k - 4$

So get $3^2 - 4k(k - 4) > 0$

$9 - 4k^2 + 16k > 0$

$4k^2 - 16k - 9 < 0$ by rearranging

$(2k + 1)(2k - 9) < 0$

See image below for a sketch

The part of the graph that has been marked is below the k-axis as this corresponds to where $4k^2 - 16k - 9$ is negative or less than zero

$- \frac{1}{2} < k < \frac{9}{2}$

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### discriminant inequalities

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