Solving For Zeroes

Content

This session intends to introduce roots (or zeroes) of functions. By mastering the content in this session, you will be able to:

Understand that a root is an value that outputs zero from a function
Algebraically solve for the roots of simple equations
Apply the guess and check method to solve for roots of more complicated functions

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Identifying Zeroes on Graphs

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Root Finding

Name

Prework

Homework

When you have some function , it is often very useful to be able to find the values of where maps to zero as shown:

Set Alt Text — Showing the mapping to zero

Remember that since on a graph, this corresponds to finding every location where the function hit’s the x-axis because:

Graphs show points , where .
Zeroes occur at , whenever

If we had the function:

There is only one value that will make it equal zero.

A Failing Case

If you tried , you will find that it does not equal zero:

As you can see does not equal 0, so is not a zero of this function.

Finding the Zero

On the other hand, we can find the zero directly by just setting and using algebra to solve for the correct value:

Now we just make the subject:

Now we cube root both sides to find x. We’ve swapped the left and right:

Now this tells us that the zero is .

Checking the Answer

We can check that by just substituting the value back into the expression, so:

So as you can see, this value is indeed a zero

Multiplying Zeroes

If we can express a function in a factorized form, we can easily solve for all of the zeroes of that function. Because of the product rule:

If can be factorized into a product of functions: Then any one of these can be true …

This is because if you have one zero in a product, the whole product must ends up being zero (). This makes it possible to solve problems when there is more than one zero.

If you had a function:

You can find its zeroes by setting:

Since in this case:

Solving a(x)

Now it’s easy to solve for . We just have:

Now making the subject gives:

Now we can test that this is truly a zero by substituting into the original function to get:

Now the first bracket here was zero, so:

This is our first zero

Solving b(x)

Now we just repeat the same thing. Remember , so we set this equal to 0 to get:

Then we make the subject again to get:

So this gives us the second zero. We can again test this, so we will see that:

So that means we have two zeroes and . Usually, we get one zero for each factor (but they can be repeated).

As we stated in a previous session, factorizing functions can become very difficult when the functions become larger. While they are small, it is reasonably easy to factorize them.

Other Values

If you want to find out when equals some other number , you can just convert this into another problem where you want to find the zero:

Now if you define , you can find when , because at those values as shown:

If you had the function

And you wanted to know when , you could set:

Now you could move everything to the left:

So here it’s as though you were solving for its zeroes. This can be done easily by making x the subject:

Testing the answer

Now you can test that does indeed make by substituting the value into the function to get:

So it produces the required y-value of 6, as expected.

Approximate Zeroes

Sometimes functions are too complicated to factorize, so we need approximate ways to find zeroes. If our function is reasonably well behaved, we can iteratively guess-and-check. The key observation is that:

If is negative and is positive (or vice versa), then there is usually some value in between them that is zero.

This does assume that the function is reasonably well behaved, so for example, here are some exceptions to this rule:

If you do have a nice function (more on what that means later), to approximate the zero, find some some left and right bounds that are positive and negative. Then keep guessing to find values in between them that are closer to zero. Repeat this until you have a precise answer:

Try this yourself. I’ve picked a random function, try to guess where it’s zero is with as few checks as you can:

0 3 3 a b x a x b CHECK RESTART Total Checks

Powers of Numbers

We know how to calculate something like by repeated multiplication, but we want to calculate something like . Since we know how to find the roots of a function, we can approximate:

Approximate the value of a surd where is a rational numbers ()

For example if you wanted to approximate

We’d start by writing 1.4 as a fraction, which is just . We then raise both sides to the power of to get:

Now we move the 128 to the left:

As you can see, the left can be written as the function:

Finding a Starting Point

We expect . So it makes sense to expect the value to be in the range . To make sure, let’s test this first:

So we can see that and . Now we will keep trying new values for x to get closer to the true value, we highlight either the minimum or maximum that changed:

Test Value	Result	Minimum	Maximum
3	f(3) = 115	2	3
2.5	f(2.5) = -30.34	2.5	3
2.7	f(2.7) = 15.49	2.5	2.7
2.6	f(2.6) = -9.19	2.6	2.7
2.63	f(2.63) = -2.17	2.63	2.7
2.65	f(2.65) = 2.69	2.63	2.65
2.64	f(2.64) = 0.24	2.63	2.64

We could continue this process, but at the end we determine that is between 2.63 and 2.64, which we write as:

Now the actual value is 2.639015822…, so we were in the correct range. We could just continue this process for as long as we like to generate more digits as we please.

Whilst this is only an approximation, remember that we can’t exactly find a rational value for a surd, that is why we call it a surd. We can however, get as many digits as we like using this method.

Solving for Zeroes

Working with Functions (6 of 12)

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