Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 3 • Question 21

Carla wants to solve the equation (𝑥 − 2)² + 𝑥 − 4 = 0 using graphical methods. She draws the graph of 𝑦 = (𝑥 − 2)² and a straight line graph on the same grid. The graph of 𝑦 = (𝑥 − 2)² has been shown on the grid below. (a) Complete Carla’s method to solve (𝑥 − 2)² + 𝑥 − 4 = 0. (b) Leon uses the following method to solve (𝑥 − 2)² − 9 = 0: (𝑥 − 2)² − 9 = 0 Add 9 to both sides, (𝑥 − 2)² = 9 Take the square root of both sides, 𝑥 − 2 = 3 Add 2 to both sides, 𝑥 = 5. Evaluate Leon’s answer and his method.

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Video Transcript

Carla wants to solve the equation 𝑥 minus two all squared plus 𝑥 minus four equals zero using graphical methods. She draws the graph of 𝑦 equals 𝑥 minus two all squared and a straight line graph on the same grid. The graph of 𝑦 equals 𝑥 minus two all squared has been shown on the grid below. Part a) Complete Carla’s method to solve 𝑥 minus two all squared plus 𝑥 minus four equals zero. There is also a part b) that we’ll come on to.

So the first thing we need to do with this problem is work out what line or straight line graph that we’re going to draw. Well, what we want to do is we want to rearrange the equation we’ve got to make 𝑥 minus two all squared on the left-hand side on its own. And then we’ll see what this equal to on the right-hand side.

So then what we’re gonna do is add four to each side of the equation. So we get 𝑥 minus two all squared plus six equals four. Then we’re gonna subtract 𝑥 from each side of the equation. And we’re gonna get 𝑥 minus two all squared is equal to four minus 𝑥.

So what this tells us is that the solutions to 𝑥 minus two all squared plus 𝑥 minus four equals zero can be found where 𝑥 minus two all squared is equal to four minus 𝑥. So therefore, the line that we want to draw is 𝑦 equals negative 𝑥 plus four. And this is the same as four minus 𝑥. All I’ve done is I’ve rearranged it just so we’ve got it in the form 𝑦 equals 𝑚𝑥 plus 𝑐.

And the reason we do this is because, as we alluded to just now, the solutions to 𝑥 minus two squared plus 𝑥 minus four equals zero will occur where 𝑦 equals 𝑥 minus two all squared, so the curve is already on the grid, and 𝑦 equals negative 𝑥 plus four meet. And that’s the straight line we’re about to draw.

So what we need to do is find two points on the line 𝑦 equals negative 𝑥 plus four so that we can join these to create the line. So the points we’re gonna choose are where it meets the 𝑥- and 𝑦-axis.

So first of all, we’re gonna start with where it meets the 𝑦-axis, start where 𝑥 is equal to zero. So we’re gonna substitute 𝑥 is equal to zero into 𝑦 equals negative 𝑥 plus four. So therefore, we get 𝑦 is equal to negative zero plus four. So we just get 𝑦 is equal to four. So our first point will be at zero, four.

Then if we get 𝑦 is equal to zero, this will be where the line crosses the 𝑥-axis. And we’re gonna have zero is equal to negative 𝑥 plus four. So therefore, if we add 𝑥 to each side of the equation, we’re gonna get 𝑥 is equal to four. So therefore, the coordinates of this point are gonna be four, zero.

So now what we do is we join these two points together to form our line. So we’ve now drawn the line 𝑦 equals negative 𝑥 plus four. So, as we said before. we want to see where this meets or intersects with the curve 𝑦 equals 𝑥 minus two all squared. Well, it’s at these two points that I’ve circled in pink.

So then to find the solutions to the equation, what we do is we read off the 𝑥-coordinates where the line and the curve meet or intersect. And we can see that there are 𝑥 equals zero and 𝑥 equals three. So therefore, we can say, using our graphical method, the solutions to the equation 𝑥 minus two all squared plus 𝑥 minus four equals zero are 𝑥 equals zero and 𝑥 equals three.

So now we can move on to part b. So part b), Leon uses the following method to solve 𝑥 minus two all squared minus nine equals zero. So we’ve got 𝑥 minus two all squared minus nine equals zero. So first, we add nine to both sides. So then we get 𝑥 minus two all squared equals nine. So then we take the square root of both sides, which gives us 𝑥 minus two equals three. So now we add two to both sides, which gives us 𝑥 equals five as the answer.

Evaluate Leon’s answer and his method.

So what we’re gonna do to evaluate his method is go through it step by step. So first of all, Leon says add nine to both sides of the equation. Well, this is correct. And if we add nine to both sides of the equation, we’re gonna get 𝑥 minus two all squared equals nine. And when we’ve done this, what we get is 𝑥 minus two all squared equals nine. So this is both correct. So now what does he do next?

Well, he takes the square root of both sides. Well, as we’ve got 𝑥 minus two all squared is equal to nine, it just looks sensible because this removed the squared from the 𝑥 minus two. So then when we do take the square root of both sides, on the left-hand side of the equation, we’re gonna be left with 𝑥 minus two, which is correct.

However, on the right-hand side of the equation, this is where the error occurs, because it’s actually only half right. That’s because it says take the square root of both sides. So if we take the square root of nine, in the answer that Leon is given, we get three. However, Leon has only found one of the solutions, because root nine is equal to positive or negative three instead of just three. So that’s gonna cause us to only find one solution, whereas we should, as I said, have two.

So now let’s carry on with his method to see how it goes. So next, he says add two to both sides. Well, again, this is correct, because that’s what we want to do to leave 𝑥 on its own on the left-hand side of the equation. So then we’ll have 𝑥 on its own on the left-hand side of the equation.

But on the right-hand side of the equation, again, we’re only half right. That’s because five is one of our answers, but we should have another answer. And that other answer should be 𝑥 is equal to negative one. That’s because if we had positive three add two, that would give the answer that we’ve got, which is five. However, if we had the other solution to root nine, which is negative three, then we’d have negative three plus two, which would give us negative one.

So what we can say as our evaluation to Leon’s answer or his method is that Leon has only found one of the solutions, because root nine is equal to positive or negative three instead of just three. So therefore, 𝑥 should and can be five or negative one.

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