Write 𝑥 squared plus eight 𝑥
minus three in the form 𝑥 plus 𝑎 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 𝑥 squared plus 𝑎𝑥, then this is gonna be equal to 𝑥 plus 𝑎 over two. So we’ve actually divided the
coefficient of our 𝑥 term by two and then this is all squared then minus 𝑎 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 𝑥
squared plus eight 𝑥 minus three is equal to 𝑥 plus eight over two all squared and
that’s because eight over two is eight is our 𝑎, is our coefficient of 𝑥. 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 𝑥
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 𝑥 plus four all squared
minus 16 minus three. So therefore, we can say that 𝑥
squared plus eight 𝑥 minus three in the form 𝑥 plus 𝑎 squared plus 𝑏 is equal to
𝑥 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 𝑥 plus four all squared
minus 19, which is gonna give us 𝑥 plus four multiplied by 𝑥 plus four minus
19. So now, we expand these
brackets. So we have 𝑥 multiplied by 𝑥
which is 𝑥 squared, 𝑥 multiplied by four which gives us four 𝑥, four multiplied
by 𝑥 which gives us another plus four 𝑥, 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 𝑥 squared plus
eight 𝑥 because we had plus four 𝑥 plus four 𝑥 so plus eight 𝑥 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 𝑥 plus 𝑎 all squared plus 𝑏, we get 𝑥 plus four all squared