First slide

Theory of equations

Question

If the roots of the quadratic equation $x^{2} - 4 x - \log_{3} a = 0$ are real, then the least value of $a$ , is

Moderate

A

81
B

1/81
C

1/64
D

none of these

Solution

Since the roots of the given equation are real
$∴$ Discriminant> 0
$\Rightarrow 16 + 4 \log_{3} a \geq 0 \Rightarrow \log_{3} a \geq - 4 \Rightarrow a \geq 3^{- 4} \Rightarrow a \geq 1 / 81$
Hence, the least value of a is 1/81.

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