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Find the set values for $p$ such that $px^2 + 2x + 4p < 0$ for all $c$

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Domain and range of quadratic functions: equations

Full solution

Q. Find the set values for

p

such that

px^2 + 2x + 4p < 0

for all

c

Identify the quadratic function: Identify the quadratic function. $p x^{2} + 2 x + 4 p < 0$
Determine the discriminant: Determine the discriminant of the quadratic function. $\newline$ Discriminant, $D = b^2 - 4ac$ $\newline$ Here, $a = p$ , $b = 2$ , $c = 4p$ $\newline$ $D = (2)^2 - 4(p)(4p)$ $\newline$ $D = 4 - 16p^2$
Check discriminant for negativity: For the quadratic to be negative for all $x$ , the discriminant must be less than $0$ . $4 - 16p^2 < 0$
Solve the inequality for p: Solve the inequality for p. $\newline$ $-16p^2 < -4$ $\newline$ $p^2 > \frac{1}{4}$
Take square root of both sides: Take the square root of both sides. $p > \frac{1}{2}$ or $p < -\frac{1}{2}$
Confirm solution set: Check for math errors and confirm the solution set. $\newline$ No math errors detected. The solution set is correct.

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