Chapter 8.6, Problem 49HP

To determine

To determine whether the absolute value of $b$ is sometimes, always or never be less than the absolute value of $c$ .

Answer to Problem 49HP

Sometimes

Explanation of Solution

Given:

  $x^2+bx+c$

Sometimes, the sum of two numbers $\left(b\right)$ is less than the product of two numbers $(c)$ except when one number is $1$ or both numbers are $2$ and $-2$ .

Therefore $\left|b\right|$ will not be less than $\left|c\right|$ except when one factor of $c$ is $1$ or if the two factors of $c$ are $2$ and $-2$ .

For example:

  $\Rightarrow x^2+5x+4=(x+1)(x+4)$ and $\left|b\right|=5>4=\left|a\right|$

  $\Rightarrow$ $x^2-4x+4=(x-2)(x+2)$and $\left|b\right|=4=\left|a\right|$

Conclusion:

Therefore, the absolute value of $b$ is sometimes less than the absolute value of $c$ .