By using regula falsi method find the real root of equation x^2+x-1 for intervals a=1 and b=2

The Regula Falsi method (also known as the False Position method) is used to find the real root of a given function within an interval. The function provided is:

\[
f(x) = x^2 + x - 1
\]

We are asked to find the root of this equation within the interval \([1, 2]\). The steps for the Regula Falsi method are as follows:

### Step 1: Check values at \(a = 1\) and \(b = 2\)

First, evaluate the function at the endpoints \(a = 1\) and \(b = 2\):

\[
f(1) = 1^2 + 1 - 1 = 1
\]
\[
f(2) = 2^2 + 2 - 1 = 5
\]

Since \(f(1) = 1\) and \(f(2) = 5\), the signs are not opposite, which suggests that the interval \([1, 2]\) does not bracket the root, but we can continue to calculate further to refine this interval. 

However, Regula Falsi needs a bracketing interval (i.e., \(f(a) \times f(b) < 0\)). Let's instead check a value between 1 and 2 to adjust the interval or refine the next point.

Would you like to check details or revise intervals? Otherwise, do

Explore the Regula Falsi method to find the real root of the quadratic equation x^2 + x - 1 within the interval [1, 2]. This solution covers key numerical methods concepts and step-by-step calculations. Suitable for students in Grades 11-12.

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