Video: Pack 1 β€’ Paper 2 β€’ Question 16

Pack 1 β€’ Paper 2 β€’ Question 16

02:22

Video Transcript

Write π‘₯ squared plus eight π‘₯ minus three in the form π‘₯ plus π‘Ž all squared plus 𝑏.

In order to actually get it into this form, what we’re gonna be doing is actually something called completing the square. So in order to complete the square, what we’ll use is this general form. If we have an expression in the form π‘₯ squared plus π‘Žπ‘₯, then this is gonna be equal to π‘₯ plus π‘Ž over two. So we’ve actually divided the coefficient of our π‘₯ term by two and then this is all squared then minus π‘Ž over two all squared. So now, let’s use this to actually complete the square for our expression that we have.

So what we’re gonna have is π‘₯ squared plus eight π‘₯ minus three is equal to π‘₯ plus eight over two all squared and that’s because eight over two is eight is our π‘Ž, is our coefficient of π‘₯. And then, we divide it by two and then minus eight over two all squared then minus three. And this is gonna be equal to π‘₯ plus four all squared and that’s because eight over two or eight divided by two is four minus four squared minus three, which will give us π‘₯ plus four all squared minus 16 minus three. So therefore, we can say that π‘₯ squared plus eight π‘₯ minus three in the form π‘₯ plus π‘Ž squared plus 𝑏 is equal to π‘₯ plus four all squared minus 19.

But what we can do is we can actually check this by expanding just to prove that it works. So we have π‘₯ plus four all squared minus 19, which is gonna give us π‘₯ plus four multiplied by π‘₯ plus four minus 19. So now, we expand these brackets. So we have π‘₯ multiplied by π‘₯ which is π‘₯ squared, π‘₯ multiplied by four which gives us four π‘₯, four multiplied by π‘₯ which gives us another plus four π‘₯, and then finally four multiplied by four which gives us plus 16, and then we already have our minus 19 on the end.

Okay, great, so now, let’s collect up our terms. So this is equal to π‘₯ squared plus eight π‘₯ because we had plus four π‘₯ plus four π‘₯ so plus eight π‘₯ and then plus 16 minus 19 gives us minus three. So great, yes, we proved that it is what we started with. So therefore, yes, we can confirm that in the form π‘₯ plus π‘Ž all squared plus 𝑏, we get π‘₯ plus four all squared minus 19.

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