Video: Pack 5 β€’ Paper 3 β€’ Question 11

Pack 5 β€’ Paper 3 β€’ Question 11

04:17

Video Transcript

Solve five minus three plus π‘₯ all squared equals zero. Give your answers correct to three significant figures.

So what I’m actually gonna do is show you a couple of methods how to solve this problem. For this first method, what I’m gonna do is gonna rearrange and solve the equation. So the first step is to actually add three plus π‘₯ all squared to each side. And when we do that, we get five is equal to three plus π‘₯ all squared.

So now the next step is to actually square-root each side of the equation. So when I do that, I actually get plus or minus the square root of five, because we see we square-rooted five and the answer could be positive or negative, is equal to three plus π‘₯. And then if I actually subtract three from each side, I get plus or minus the square root of five minus three is equal to π‘₯. So therefore, we get the possible answers for π‘₯. So π‘₯ will be equal to root five minus three or negative root five minus three.

If we check back and look at the question what it wants, it says that it wants it to three significant figures. So therefore, we get π‘₯ is equal to negative 0.764 or negative 5.24. And there we have it. We’ve actually solved five minus three plus π‘₯ all squared equals zero to three significant figures. And it gives us the π‘₯ values of negative 0.764 or negative 5.24.

Okay, so that’s the first method. And it’s the method I’ll probably use to solve this kind of problem. However, I’m just gonna give you an alternative so you can check it as well. And I’m gonna do that using the quadratic formula. And the quadratic formula says that if we have a quadratic in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 equals zero, then π‘₯ is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four π‘Žπ‘ over two π‘Ž.

Okay, so now, great, we’ve got a quadratic formula. Let’s use it to actually check and find the answer to this equation. So we’ve got five minus. And then we’ve got three plus π‘₯ all squared equals zero. So first of all, what we need to do is actually expand the brackets. So what we have is five minus three plus π‘₯ multiplied by three plus π‘₯. So there are two brackets. And that’s because, actually, we had three plus π‘₯ all squared. And this is equal to zero.

So we get five minus. And then it’s three multiplied by three, which is nine, and then three multiplied by π‘₯. That gives us plus three π‘₯. And then we have π‘₯ multiplied by three. So we’ve got plus another three π‘₯. And then, finally, we’ve got π‘₯ multiplied by π‘₯. That gives us π‘₯ squared. So we’ve got plus π‘₯ squared.

So we now have five minus and then in brackets nine plus three π‘₯ plus three π‘₯ plus π‘₯ squared equals zero. So then what we have is five minus nine β€” because remember we’ve got a minus in front of our brackets, so it’s like multiplying each term by minus one β€” minus six π‘₯, cause again we have plus three π‘₯ plus three π‘₯, which is plus six π‘₯, multiplied by minus one is minus six π‘₯ and then minus π‘₯ squared is equal to zero.

So then if we actually multiply each side of our equation by negative one, we get π‘₯ squared plus six π‘₯ plus four is equal to zero. Okay, great! So we’ve now got it in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 is equal to zero. So we can use the quadratic formula. So we’ve got π‘Ž is equal to one. 𝑏 is equal to six. And 𝑐 is equal to four.

So therefore, if we substituted into the quadratic formula, we get π‘₯ is equal to negative six plus or minus the square root of six squared minus four multiplied by one multiplied by four. And this is all divided by two multiplied by one. So therefore, we can say that π‘₯ is equal to negative six plus root 20 over two, or negative six minus root 20 over two.

So if you actually work this out on the calculator, you get π‘₯ is equal to negative 0.764 or π‘₯ is equal to negative 5.24. And both of these have been rounded to three significant figures. And if we check back our previous answer from the first way that we actually used to work this out, we get the same answers. So great! We can definitely say that the solution to five minus three plus π‘₯ all squared equals zero is π‘₯ is equal to negative 0.764 or π‘₯ is equal to negative 5.24.

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