Lesson video

Hello.

My name is Mr. Clasper.

And today we are going to be using a change of sign to establish where a solution lies.

How would you solve each of these equations? Well, the equation on the left, we have 7x plus three is equal to 2x plus 13.

What we could do is subtract 2x from both sides which leaves us with 5x plus three equals 13.

And from here, we can find out that 5x is equal to 10 and X is equal to two.

If you look at our second equation, we have a quadratic equation in this instance, as the highest power is a power of two.

So for this equation, we could make it equal zero by subtracting 14 from both sides and then we could factorise.

This means that we have two solutions which are X is equal to negative seven and X is equal to two.

What about this equation? How would you solve this equation? Well, we could make this equation equals zero as well.

However, as this is a cubic equation, it's not quite as straightforward as solving a quadratic equation.

Let's have a look at this equation on a graph.

This is the graph of Y equals X cubed plus 2x squared minus 30.

And this horizontal line represents the graph Y is equal to zero.

Where these two graphs meet would tell us the solution to the equation X cubed plus 2x squared minus 30 is equal to zero.

We can see from the graph that there is a solution between where X is equal to two and where X is equal to three.

This is because our original graph crosses the line Y equals zero.

When X is equal to two, the value is negative 14.

And when X is equal to three, the value is 15.

Because there has been a change in sign between going from negative 14 to positive 15, this indicates that there must be a solution between these two points.

In our example, we're going to show that there was a solution which lies between the points where X is equal to three and where X is equal to four.

This is the equation we're going to use.

If we substitute X is equal to three into the expression on the left-hand side, we return a value of negative one.

When we substitute X is equal to four, we return a value of 26.

Because our returned values were negative one and positive 26 and there is a change of sign, this means that there must be a solution between these two points.

So when we find a change of sign, we can find integer values which solutions lie between.

Here's a table of values for the same graph.

On this same graph, we also know that there must be a solution which lies between negative four and negative three.

This is because when substituted these return values of negative 22 and positive five, there is a change of sign, therefore there must be a solution.

This happens again between the values of zero and one as our returned values are two and negative seven.

So again, a change of sign would verify that we must have a solution between these two points.

This is the graph of Y equals X cubed minus 10x plus two.

We can see from the graph that we do indeed have two solutions between negative four and negative three.

We also have another solution between zero and one.

And we have another solution between three and four.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So for part A, we can see that our graph intersects the x-axis between the values of zero and one.

So this means that there must be a solution between zero and one.

And for part B, we can see that there is another solution, as the graph also intersects the x-axis between the values of negative three and negative four.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So for part A, we needed to substitute our values.

And we found that we had a negative value returned and a positive value returned.

Therefore, there must be a solution, because there is a change in sign.

If we look at part B, we need to compare the return value to the number one.

So because we get a value greater than one and less than one, this again, verifies that there must be a solution between X is equal to one and X is equal to two.

And for part C our answer was the same, or our conclusion was the same.

This is because we've been solving the same equation.

So the equation from part B is simply rearranged from part A.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So like the last examples, we're going to substitute our values into each of the given equations.

And as long as we find a positive value and a negative value, we can verify that there must be a solution between the two given points.

And that brings us to the end of our lesson.

So we've been using a change of sign to verify for solution lies between two given points.

Why not try the exit quiz to show off your new skills? I'll hopefully see you soon.