Question

If the zeroes of the quadratic polynomial ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$, where $\mathrm{c}\ne 0$, are equal, then

• A. $\mathrm{c}$ and $\mathrm{a}$ have opposite sign
• B. $\mathrm{c}$ and $\mathrm{b}$ have opposite sign
• C. $\mathrm{c}$ and $\mathrm{a}$ have the same sign
• D. $\mathrm{c}$ and $\mathrm{b}$ have the same sign

Answer 1

The correct option is C

$\mathrm{c}$ and $\mathrm{a}$ have the same sign

Basic structure of the quadratic polynomial.

As we know that a polynomial of the form $\mathrm{p}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$, where $\mathrm{a}\ne 0$ is called as a quadratic polynomial.

Where, $\mathrm{a}$ is the coefficient of ${\mathrm{x}}^2$, $\mathrm{b}$ is the coefficient of $\mathrm{x}$ and $\mathrm{c}$ is constant.

Following are the methods to determine the zeroes of quadratic polynomial,

$\mathrm{p}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$

$\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$

Here, ${\mathrm{b}}^2-4\mathrm{ac}$ are discriminant.

From the given, the zeroes of the quadratic polynomial ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ , where $\mathrm{c}\ne 0$are equal.

As we all know, if the roots or zeroes are equal, the discriminant value must be zero.

Therefore,

$\mathrm{Discriminant}={\mathrm{b}}^2-4\mathrm{ac}$

Taking $\mathrm{D}=0$, we get

$\Rightarrow$ ${\mathrm{b}}^2-4\mathrm{ac}=0$

$\Rightarrow$ ${\mathrm{b}}^2=4\times \mathrm{a}\times \mathrm{c}$

As we can see, the value of ${\mathrm{b}}^2$on the left hand side cannot be negative, because the square of a number is always positive, since, the value $4\times \mathrm{a}\times \mathrm{c}$ on the right hand side can also never be negative.

Hence, $\mathrm{a}$ and $\mathrm{c}$ have the same signs.

Therefore, option $\mathrm{C}$ is correct answer.