Question

If the roots of the quadratic equation ${x^2} - 4x - {\log _3}a = 0$ are real then the least value of $a$ is
1. $81$
2. $\dfrac{1}{{81}}$
3. $\dfrac{1}{{64}}$
4. None of these

Answer 1

In the given question we need to find the least value of the constant “a” in the given quadratic equation. In order to find this we need to find the value of the discriminant and solve it according to the condition given on the roots of the given quadratic equation.
The Discriminant Formula in the quadratic equation $a{x^2} + bx + c = 0$ is ${b^2} - 4ac$. By knowing the value of a determinant, the nature of roots can be determined as follows:
$\bullet$ If the discriminant value is positive, the quadratic equation has two real and distinct solutions.
$\bullet$ If the discriminant value is zero, the quadratic equation has only one solution or two real and equal solutions.
$\bullet$ If the discriminant value is negative, the quadratic equation has no real solutions.

Complete step-by-step solution:
We know that any quadratic equation in the variable $x$ is of the form $a{x^2} + bx + c = 0$ .
To find the values of roots of the given quadratic equation we use the quadratic formula which is as follows:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Since the roots of the given equation ${x^2} - 4x - {\log _3}a = 0$ are real, therefore the
discriminant $> 0$
i.e. ${b^2} - 4ac > 0$
Therefore ${(4)^2} - 4(1)( - {\log _3}a) > 0$
Which simplifies to
$16 + 4{\log _3}a > 0$
Therefore ${\log _3}a > - 4$
Which gives us $\dfrac{{\log a}}{{\log 3}} > - 4$
Which further implies $\log a > \log {3^{ - 4}}$
Therefore $a > {3^{ - 4}}$
And hence we get $a > \dfrac{1}{{81}}$
Therefore option (2) is the correct answer.

Note: Keep in mind the general form of a quadratic equation. Take into account the properties of the discriminant and then wisely use the conditions as per the requirement of the question. While solving the inequality, take care of the signs. Do the calculations very carefully and recheck them so as to get the correct solution.