Any equation can be solved using this method. This is how it works.
Example: \( 7x^2 + 3x + 2\cos{2x} = 5 \)
Step 1: Rearrange the equation into the form \( x = f(x) \). \( f(x) \) must be a convergent equation, meaning that it must usually return a value smaller than \( x \).
$$3x^3 + 12x^2 - x = 4$$
$$3x^3 = 4 + x - 12x^2$$
$$x^3 = \frac{4 + x - 12x^2}{3}$$
$$x = \sqrt[3]{\frac{4 + x - 12x^2}{3}}$$
Step 2: Pick a number to start. If no range is given, select any number you like. In this case, we'll use the number 0.5
$$x = \sqrt[3]{\frac{4 + 0.5 - 12(0.5)^2}{3}}$$
$$x = 0.7937$$
Step 3: Substitute the answer back into the equation. Keep doing this until the equation gives you the same number as the one you put in. In this case, we want our answer to be accurate to three significant figures. So we keep doing Step 3 until our answer (rounded to three s.f.) becomes the same.
$$x = \sqrt[3]{\frac{4 + 0.7937 - 12(0.7937)^2}{3}}$$
$$x = -0.9733$$
After many steps we get \( x = -4.00 \).