1. We are given the problem to find the root of the function $f(x) = x^2 - 3$ in the interval $[1,2]$ using the bisection method with an error tolerance $\epsilon = 0.01$. 2. The bisection method works by repeatedly halving the interval and selecting the subinterval where the function changes sign, thus narrowing down the root's location. 3. First, evaluate $f(1) = 1^2 - 3 = -2$ and $f(2) = 2^2 - 3 = 1$. Since $f(1) < 0$ and $f(2) > 0$, there is a root between 1 and 2. 4. Calculate midpoint $m = \frac{1 + 2}{2} = 1.5$. Evaluate $f(1.5) = 1.5^2 - 3 = 2.25 - 3 = -0.75$. 5. Since $f(1.5) < 0$ and $f(2) > 0$, the root is between 1.5 and 2. Update interval to $[1.5, 2]$. 6. Calculate new midpoint $m = \frac{1.5 + 2}{2} = 1.75$. Evaluate $f(1.75) = 1.75^2 - 3 = 3.0625 - 3 = 0.0625$. 7. Now $f(1.5) < 0$ and $f(1.75) > 0$, so root is between 1.5 and 1.75. Update interval to $[1.5, 1.75]$. 8. Calculate midpoint $m = \frac{1.5 + 1.75}{2} = 1.625$. Evaluate $f(1.625) = 1.625^2 - 3 = 2.640625 - 3 = -0.359375$. 9. Since $f(1.625) < 0$ and $f(1.75) > 0$, root is between 1.625 and 1.75. Update interval to $[1.625, 1.75]$. 10. Calculate midpoint $m = \frac{1.625 + 1.75}{2} = 1.6875$. Evaluate $f(1.6875) = 1.6875^2 - 3 = 2.84765625 - 3 = -0.15234375$. 11. Root lies between $[1.6875, 1.75]$ since signs differ. Calculate new midpoint $m = 1.71875$, evaluate $f(1.71875) = 2.9560546875 - 3 = -0.0439453125$. 12. Root lies between $[1.71875, 1.75]$, midpoint $m = 1.734375$, $f(1.734375) = 3.009765625 - 3 = 0.009765625$. 13. Because $f(1.71875) < 0$ and $f(1.734375) > 0$, root is in between. 14. The interval length is $1.734375 - 1.71875 = 0.015625$, which is greater than $ e 0.01$, so continue. 15. Calculate midpoint $m = 1.7265625$, $f(1.7265625) = 2.98291015625 - 3 = -0.01708984375$. 16. Root lies in $[1.7265625, 1.734375]$, midpoint $m = 1.73046875$, $f(1.73046875) = 2.996337890625 - 3 = -0.003662109375$. 17. Root in $[1.73046875, 1.734375]$, interval length $0.00390625$ which is less than $0.01$. 18. We stop here and take the approximate root as the midpoint of this interval, $$x \approx \frac{1.73046875 + 1.734375}{2} = 1.732421875$$ This is the root approximation to within the error tolerance. Final answer: $x \approx 1.732$