Question

If the roots of the quadratic equation \[{x^2} - 4x - {\log _3}a = 0\] are real, then the least value of a is
a. 81
b. \[\dfrac{1}{{81}}\]
c. \[\dfrac{1}{{64}}\]
d. 4

Answer 1

Hint: Here in this question, we have to find the value of a. The quadratic equation can be solved and determine the roots for the equation. The roots are which of the form is checked by calculating the value of discriminant. hence, we obtain the required solution for the given question.

Complete step by step solution:
The quadratic equation is an equation which contains both variables and constants. By solving the quadratic equation, we get two roots for the equation. So we can say that for the quadratic equation has the highest degree 2.
In generally the quadratic equation is defined as \[a{x^2} + bx + c\], where a, b and c are constants and x is variable.
The discriminant determines the nature of the roots of a quadratic equation. The word 'nature' refers to the types of numbers the roots can be namely real, rational, irrational or imaginary.
If the discriminant \[{b^2} - 4ac = 0\], then the roots are equal and real
If the discriminant \[{b^2} - 4ac > 0\], then the roots are real
If the discriminant \[{b^2} - 4ac < 0\], then the roots are imaginary
The roots of the quadratic equation \[{x^2} - 4x - {\log _3}a = 0\] are real, the discriminant \[{b^2} - 4ac \geqslant 0\], then the roots are real
Therefore the value of a is 1, the value of b is -4 and the value of c is \[{\log _3}a\]
so we have
\[ \Rightarrow {( - 4)^2} - 4(1)({\log _3}a) \geqslant 0\]
\[ \Rightarrow 16 - 4({\log _3}a) \geqslant 0\]
Take 16 to RHS of the equation
\[ \Rightarrow - 4({\log _3}a) \geqslant - 16\]
Divide the above equation by -4 we have
\[ \Rightarrow {\log _3}a \leqslant 4\]
Then we have
\[ \Rightarrow a = {3^4}\]
On multiplying the number 3 to 4 times we have
\[ \Rightarrow a = 81\]
Therefore the least value of a is 81.

So, the correct answer is “Option A”.

Note: The equation is a quadratic equation. This defines as for the general quadratic equation \[a{x^2} + bx + c\], by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] we can determine the roots for the equation. But the nature of roots are determined by the formula \[{b^2} - 4ac\], and we can say the roots are real or equal or imaginary.