mbs1

Algebra Level 3

$f(x)=ax^2+bx+c$ is a quadratic polynomial with a, b, and c as real numbers. $\alpha$ and $\beta$ are real and roots of $f(x)$ . It is given that $\alpha < -1$ and $\beta > 1$ . What is the sign of the expression $1+\frac{c}{a}+|\frac{b}{a}|$

Negative Positive

1 solution

X X
Jul 28, 2018

This expression $=1+\alpha\beta+|\alpha+\beta|$

If $\alpha+\beta>0$ ,then it becomes $(1+\alpha)(1+\beta)$ which is negative.

If $\alpha+\beta<0$ ,then it becomes $(1-\alpha)(1-\beta)$ which is also negative.

mbs1

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