Question

If one of the zeroes of a quadratic polynomial of the form ${{x}^{2}}+ax+b$ is the negative of the other, then it
(a) has no linear term and the constant term is negative.
(b) has no linear term and the constant term is positive.
(c) can have a linear term but the constant term is negative.
(d) can have a linear term but the constant term is positive.

Answer 1

Hint: Write the quadratic equation in terms of roots. So by this we get the relation between the roots and the coefficients. We say that the coefficients can be said as sum of roots and product of roots.

Complete step-by-step answer:
If p, q are roots of $a{{x}^{2}}+by+c=0$ Then
$p+q=-\dfrac{b}{a}$ and $pq=c$
Mathematically we say that:
p, q are roots of ${{x}^{2}}-\left( p+q \right)x+pq=0$
Given in question that $p=-q$
By substituting this we turn the equation into:
${{x}^{2}}-pq=0$
So, the equation has no linear term and constant term of the equation is negative.
Therefore option(a) is correct.

Note: Be careful while converting the equation into terms of its roots. In this case p, q will cancel, but if not then the minus sign on the linear term is very important. As it is a term defining the slope also.