Question

If the roots of the equation $a{x^2} + bx + c = 0$ are imaginary, then for all values of a, b, c and $x \in {\mathbf{R}}$, the expression ${a^2}{x^2} + abx + ac$ is
A) Positive
B) Non – negative
C) Negative
D) May be positive, zero and negative

Answer 1

Hint:
The discriminant D, given by $D = {b^2} - 4ac$, of the equation $a{x^2} + bx + c = 0$ will be less than zero since its roots are imaginary. Using this condition, we will calculate the discriminant of the second equation ${a^2}{x^2} + abx + ac$ to determine its nature.

Complete step by step solution:
We are given that roots of the equation $a{x^2} + bx + c = 0$ are imaginary for all values of a, b, c and $x \in {\mathbf{R}}$.
The discriminant of this equation will be given by the formula: $D = {b^2} - 4ac$ will be
\[ \Rightarrow D = {b^2} - 4ac\] and since the roots of the equation are imaginary, hence, D < 0 i.e. ${b^2} - 4ac < 0$
Now, for the second equation ${a^2}{x^2} + abx + ac$, we will calculate the discriminant again using the same formula $D = {b^2} - 4ac$, we get
$ \Rightarrow D = {\left( {ab} \right)^2} - 4\left( {{a^2}} \right)\left( {ac} \right)$
$ \Rightarrow D = {a^2}{b^2} - 4{a^3}c = {a^2}\left( {{b^2} - 4ac} \right)$
Using the condition that ${b^2} - 4ac < 0$ in the above equation, we get
$ \Rightarrow D = {a^2}\left( {{b^2} - 4ac} \right) < 0$
Now, since the discriminant of this equation is less than zero, the roots will be imaginary and $a \ne 0$. But, as we know that the sign of the coefficient of ${x^2}$ is the sign of the expression i.e. if the coefficient of ${x^2}$is positive, then the equation will be positive and vice – versa.
Since $a \ne 0$ can take any value but the square of “a” will always be positive, therefore, we can write that the sign of the coefficient of ${x^2}$ will be positive and hence the equation can be written as
$\therefore {a^2}{x^2} + abx + ac > 0$

Therefore, the given equation is found to be positive.
Hence, option (A) is correct.

Note:
In such questions, you may get confused after calculating the D of the first equation in how to proceed with. For the same first we need to compare the given quadratic equation to the standard quadratic equation then we’ll get the value of coefficients. Then, we can directly use formulas.