Write 𝑥 squared plus six 𝑥 minus
two in the form 𝑥 plus 𝑎 all squared plus 𝑏 where 𝑎 and 𝑏 are integers.
Writing a quadratic expression in
this form is called completing the square. As with factorizing, there are lots
of methods to get to the correct answer. In this case, we will look at
Our first method will be a
step-by-step process that will work every time the coefficient of 𝑥 squared is
equal to one. This means there is no number in
front of the 𝑥 squared term. Our first step using this method is
to divide the coefficient of 𝑥 by two. In this case, six divided by two is
equal to three. Our next step is to subtract the
square of the number we have just written inside the bracket. In this case, we’re going to
subtract three squared. Finally, we dropped the last term
into the next line, in this case negative two.
Simplifying this gives us 𝑥 plus
three all squared minus nine minus two. And finally, grouping the last two
terms gives us 𝑥 plus three all squared minus 11. Our expression has now been written
in the correct form, where 𝑎 is equal to three and 𝑏 is equal to negative 11.
An alternative method that you
might have been taught is that any quadratic expression 𝑎𝑥 squared plus 𝑏𝑥 plus
𝑐 can be rewritten as 𝑎 multiplied by 𝑥 plus 𝑝 all squared plus 𝑞, where 𝑝 is
equal to 𝑏 divided by two 𝑎 and 𝑞 is equal to 𝑐 minus 𝑏 squared divided by four
𝑎. In our expression 𝑥 squared plus
six 𝑥 minus two, our values of 𝑎, 𝑏, and 𝑐 are one, six, and negative two,
respectively. 𝑝 can, therefore, be calculated by
dividing six by two multiplied by one. This is equal to three.
𝑞 is equal to negative two minus
six squared divided by four multiplied by one. Six squared is equal to 36 and four
multiplied by one is equal to four. 36 divided by four is equal to
nine, which leaves us with 𝑞 is equal to negative two minus nine. This is equal to negative 11. As 𝑎 is equal to one, substituting
in our values of 𝑝 and 𝑞 gives us the same answer.
𝑥 squared plus six 𝑥 minus two
can be rewritten as 𝑥 plus three all squared minus 11.