Question

Find the real root of the equation x log x -1.2 = 0 by regula falsi method correct to three
decimal places.

Answer 1

Regula Falsi Method:

Step-by-step explanation:

Suppose $f(x)=xlogx-1.2$

Then, $f(2)=2log2-1.2=-0.5979$

And, $f(3)=3log3-1.2=0.2314$

Since, $f(2)$ and $f(3)$ are of opposite signs, the real roots lies between $x_{1}=2$ and $x_{2}=3$.

Therefore, the first approximation is obtained from $x_{3}=x_{2}-\frac{x_{2}-x_{1}}{f(x_{2})-f(x_{1})} f(x_{2})$

That means,

$x_{3}=3-\frac{3-2}{0.2314-(-0.5979)} (0.2314)\\x_{3}=3-\frac{0.2314}{0.2314+0.5979} \\x_{3}=3-\frac{0.2314}{0.8293} \\x_{3}=2.72097$

Therefore, $f(x_{3})=2.72097log2.72097-1.2=-0.01713$

Since, $f(x_{2})$ and $f(x_{3})$ are of opposite signs, the real roots lies between $x_{2}$ and $x_{3}$.

Now, the second approximation is given by $x_{4}=x_{3}-\frac{x_{3}-x_{2}}{f(x_{3})-f(x_{2})} f(x_{3})$

That means,

$x_{4}=2.72097-\frac{2.72097-3}{-0.01713-0.2314} (-0.01713)\\x_{4}=2.72097-(\frac{-0.27903}{-0.24853}) (-0.01713)\\x_{4}=2.72097-(-0.01923)\\x_{4}=2.7402$

Therefore, $f(x_{4})=2.7402log2.7402-1.2=-0.0038905$

Thus, the real root of the given equation, $xlogx-1.2=0$ correct to three decimal places is $2.740$