Question

If the zeroes of the quadratic polynomial $a{{x}^{2}}+bx+c,c\ne 0$ are equal, then
A. c and a have opposite signs
B. c and b have opposite signs
C. c and a have the same sign
D. c and b have the same sign

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:
$f(x)=a{{x}^{2}}+bx+c,\quad a\ne 0$
The discriminant of the polynomial could be stated as: $D={{b}^{2}}-4ac$.
As per given in the question, the zeroes of the polynomial $a{{x}^{2}}+bx+c,c\ne 0$ are equal.
If the roots are equal, then the value of discriminant has to be zero.
The formula of discriminant is: $D={{b}^{2}}-4\times a\times c$
  Now, putting D = 0, we get
\[\begin{align}
  & \Rightarrow {{b}^{2}}-4\times a\times c=0 \\
 & \Rightarrow {{b}^{2}}=4\times a\times c \\
\end{align}\]
As we know that the value of ${{b}^{2}}$(Left hand side) cannot be negative as the square of a number is always positive, thus $4\times a\times c$ (Right hand side) can also never be negative.
Hence, the a and c must be of the same sign.
Therefore, option (c) is correct.

Note: This problem could be alternatively solved by using the relationship between zeroes and coefficients of quadratic polynomials. As we know that the product of zeros of a quadratic polynomial is equal to $\dfrac{c}{a}$. Since both the zeros are equal, the product becomes a square. So, we obtain the same relationship as obtained above.