Cubic Log

Algebra Level 3

Find the real value of $x$ satisfying the equation below.

$\log_{10} (98 + \sqrt{x^3 - x^2 - 12x +36}) = 2$

1 solution

Rishik Jain
Jan 5, 2016

$\because log_{10}100=2$

$98 + \sqrt{x^3 - x^2 - 12x +36} = 100$

$\sqrt{x^3 - x^2 - 12x +36} = 2$

$x^3 - x^2 - 12x +36 = 4$

$x^3 - x^2 - 12x +32 = 0$

Solving this cubic equation, we get only 1 real value of $x$ ,

$\large \boxed{x = -4}$

By the way, which method was used to solve the cubic equation ?

Akshat Sharda - 5 years, 5 months ago

The equation had small integral coefficients. There was no need to use Cardano's Method here. A simple hit and trial would suffice.

Rishik Jain - 5 years, 5 months ago

It can be easily done using trial ,I don't think we should use Cardan's here.

Rohit Udaiwal - 5 years, 5 months ago

Factor theorem

Deepak Kumar - 5 years, 5 months ago