Let's say we have the quintic we want to find the zero for:

We can guess a low and high x value for example looking at the graph we might pick 1.4 and 1.5...

We find the value of the function at those x positions:

Now we can find a quadratic equation going through these above two points and a point between them at x=r whose value is 0, by solving 3 equations for the unknown coefficients. Here I'll just use Maple's interpolate command to find it...

This looks like a mess, but here we apply the idea that the closest quadratic equation to the quintic we are finding the 0 for will have to look like the quintic equation truncated to the x^2, x, and constant terms. In particular this means we can look at just the constant term of the quintic which happened to be 1 and the last term of this quadratic we just found must also have a constant term of 1. So set those equal and solve:

There might might be two roots but we want the one that is between our chosen points of 1.4 and 1.5, 1.472384325.

Now we want to evaluate the quintic at this value and see whether it is negative or positive so we know which side of the root it lies on....

Now we iterate the above steps using this new point as our new left value:

First interpolating the 3 points to find the quadratic...

Then setting the constant term of this quadratic equal to the constant term of the equation we're trying to find the root of and solving for r:

Then either accepting this as close enough to the root or continue iterating to get ever closer.

I don't believe this is the method the computer uses but it has other ways...

Our series was 1.4, 1.472484325, 1.475958045... We were closing in fast to the approximation the computer gives of 1.476126271...

There are a lot of other ways to find roots but I thought this one was interesting.

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