Question

By assuming that the solution to the equation $x^4 - x - 10 = 0$ lies between $x = 1.8$ and $x = 2$, then the next approximate solution by Regula-falsi method is (Take $f(1.8) = -1.3024$):

• A. $1.8491$
• B. $1.9932$
• C. $1.8001$
• D. $1.9091$

Hint:

Regula-Falsi uses the formula of a straight line connecting $(a, f(a))$ and $(b, f(b))$. Keep track of the negative signs carefully in the denominator, as subtracting a negative number yields an addition!

Answer 1

The Correct Option is A

Concept: The Regula-Falsi (False Position) method is an iterative method for finding roots of a non-linear equation $f(x) = 0$. Given an interval $[a, b]$ such that $f(a) \cdot f(b) < 0$, the formula for the next approximation $x_1$ is given by: \[ x_1 = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)} \] Here, the function is $f(x) = x^4 - x - 10$. The interval boundaries are $a = 1.8$ and $b = 2$.

Step 1: Calculate the function values at the endpoints.
We are given: \[ f(a) = f(1.8) = -1.3024 \] Now, let us manually evaluate $f(b) = f(2)$: \[ f(2) = 2^4 - 2 - 10 = 16 - 2 - 10 = 4 \] Since $f(1.8) = -1.3024$ (negative) and $f(2) = 4$ (positive), a root is guaranteed to lie in this interval.

Step 2: Substitute these values into the Regula-Falsi formula.
\[ x_1 = \frac{(1.8) \cdot f(2) - (2) \cdot f(1.8)}{f(2) - f(1.8)} \] Substitute $f(2) = 4$ and $f(1.8) = -1.3024$: \[ x_1 = \frac{(1.8)(4) - (2)(-1.3024)}{4 - (-1.3024)} \] Calculate the numerator: \[ (1.8)(4) = 7.2 \] \[ -(2)(-1.3024) = +2.6048 \] \[ \text{Numerator} = 7.2 + 2.6048 = 9.8048 \] Calculate the denominator: \[ \text{Denominator} = 4 + 1.3024 = 5.3024 \]

Step 3: Perform the division to find $x_1$.
\[ x_1 = \frac{9.8048}{5.3024} \approx 1.84912 \] Rounding to four decimal places gives $1.8491$.