Question

The zeros of the quadratic polynomial ${{x}^{2}}+99x+127=0$ are
A.Both positive
B.Both negative
C.Both equal
D.One positive and one negative

Answer 1

Hint: For solving this problem, we obtain the sum and product of roots individually from the given equation and then place the values of variables in the generalized form to obtain a relationship between zeros of polynomials.

Complete Step-by-step answer:
In algebra, a quadratic function is a polynomial function with one or more variables in which the highest-degree term is of the second degree. A single-variable quadratic function can be stated as:
$f(x)=a{{x}^{2}}+bx+c,\quad a\ne 0$
If we have two zeros of a quadratic equation then the polynomial could be formed by using the simplified result which could be stated as:
${{x}^{2}}-(\alpha +\beta )x+\alpha \beta $, where $\alpha \text{ and }\beta $ are two zeroes of the equation.
According to the problem statement, we are given a quadratic polynomial ${{x}^{2}}+99x+127=0$.
By using the above expression, the sum of zeroes could be expressed as
\[\begin{align}
  & -(\alpha +\beta )=99 \\
 & \alpha +\beta =-99 \\
\end{align}\]
The product of zeroes could be expressed
$\alpha \beta =127$
From the above statement, we conclude that the sum of zeroes is negative while the product of zeros is positive. This is only possible when both of the zeroes are negative.
Therefore option (b) is correct.

Note: By using the discriminant formula i.e. $D=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$, we can evaluate the respective values of roots by putting a = 1, b = 99 and c = 127 and conclusively determine whether the roots are positive, negative or changing sign(one positive and other one negative).