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+# Root Finding Methods
+For linear equations, finding the root is straightforward—you can usually solve for the unknown just by rearranging the equation algebraically. With nonlinear equations, however, the process quickly becomes more challenging.
+
+Up to this point, you've already learned tools like the quadratic formula,
+$$
+x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}
+$$
+this works nicely if we're given the function $f(x)$ of a second degree polynomial. However, as engineers we need to be able to solve equations that do not follow this trend, or perhaps we are working with experimental data rather than an analytical function. In this section we will cover bracketing and open methods to find roots of a function.
+
+Let us take a step back and think *algorithmically*. To find a root numerically, we can implement a search algorithm. Let's consider a small phone book - a database of names and numbers sorted alphabetically by last name. Our goal is to find the phone number of James Watt (our root). We could make an algorithm that starts at the the first page, checks to see if the current name equals our target (Watt), if it doesn't, check the next entry. This is method is called *incremental search*. As you may see, this can take very long if the database is large.
+
+> [!NOTE] Example Phonebook Database Entries
+> **A** – Archimedes  
+> **B** – Bernoulli
+> **C** – Curie
+> **D** – Darwin
+> **E** – Einstein
+> **F** – Feynman
+> **G** – Galileo
+> **H** – Hubble
+> **I** – Ibn Sina
+> **J** – Joule
+> **K** – Kelvin
+> **L** – Leonardo
+> **M** – Maxwell
+> **N** – Newton 
+> **O** – Oppenheimer 
+> **P** – Pascal
+> **Q** – Quetelet
+> **R** – Röntgen
+> **S** – Schrödinger
+> **U** – Urey
+> **V** – Volta
+> **W** – Watt
+> **X** – Xenophon 
+> **Y** – Yukawa
+> **Z** – Zuse
+
+Another approach may be to index the book and jump straight to the 'W' names however this requires two steps, we first need to. How would you approach this problem?
+
+[[2_1_Bracketing_Method]]
+[[2_2_Open_Methods]]
+[[2_3_Systems_of_Non-linear_equations]]
+\ No newline at end of file