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# Worked example: completing the square (leading coefficient ≠ 1)

CCSS Math: HSA.REI.B.4, HSA.REI.B.4a, HSA.REI.B.4b, HSA.SSE.B.3, HSA.SSE.B.3b, HSF.IF.C.8, HSF.IF.C.8a

## Video transcript

We're asked to
complete the square to solve 4x squared plus
40x minus 300 is equal to 0. So let me just rewrite it. So 4x squared plus 40x
minus 300 is equal to 0. So just as a first
step here, I don't like having this 4 out
front as a coefficient on the x squared term. I'd prefer if that was a 1. So let's just divide both
sides of this equation by 4. So let's just divide
everything by 4. So this divided
by 4, this divided by 4, that divided by 4,
and the 0 divided by 4. Just dividing both sides by 4. So this will simplify
to x squared plus 10x. And I can obviously
do that, because as long as whatever I do
to the left hand side, I also do the right
hand side, that will make the equality
continue to be valid. So that's why I can do that. So 40 divided by 4 is 10x. And then 300 divided
by 4 is what? That is 75. Let me verify that. 4 goes into 30 seven times. 7 times 4 is 28. You subtract, you
get a remainder of 2. Bring down the 0. 4 goes into 20 five times. 5 times 4 is 20. Subtract zero. So it goes 75 times. This is minus 75 is equal to 0. And right when you look at
this, just the way it's written, you might try to factor
this in some way. But it's pretty clear this
is not a complete square, or this is not a perfect
square trinomial. Because if you look at
this term right here, this 10, half of this 10 is 5. And 5 squared is not 75. So this is not a perfect square. So what we want to
do is somehow turn whatever we have on
the left hand side into a perfect square. And I'm going to start out by
kind of getting this 75 out of the way. You'll sometimes
see it where people leave the 75 on
the left hand side. I'm going to put on
the right hand side just so it kind of clears
things up a little bit. So let's add 75 to both
sides to get rid of the 75 from the left hand
side of the equation. And so we get x squared plus
10x, and then negative 75 plus 75. Those guys cancel out. And I'm going to
leave some space here, because we're going
to add something here to complete the square
that is equal to 75. So all I did is add 75 to
both sides of this equation. Now, in this step,
this is really the meat of
completing the square. I want to add something to
both sides of this equation. I can't add to only one
side of the equation. So I want to add something to
both sides of this equation so that this left hand side
becomes a perfect square. And the way we can do that,
and saw this in the last video where we constructed a
perfect square trinomial, is that this last
term-- or I should say, what we see on
the left hand side, not the last term, this expression
on the left hand side, it will be a perfect square if
we have a constant term that is the square of half of the
coefficient on the first degree term. So the coefficient here is 10. Half of 10 is 5. 5 squared is 25. So I'm going to add 25
to the left hand side. And of course, in order to
maintain the equality, anything I do the left hand
side, I also have to do to the right hand side. And now we see that this
is a perfect square. We say, hey, what two numbers
if I add them I get 10 and when I multiply
them I get 25? Well, that's 5 and 5. So when we factor this, what
we see on the left hand side simplifies to, this
is x plus 5 squared. x plus 5 times x plus 5. And you can look at
the videos on factoring if you find that confusing. Or you could look
at the last video on constructing perfect
square trinomials. I encourage you to
square this and see that you get exactly this. And this will be equal to 75
plus 25, which is equal to 100. And so now we're saying
that something squared is equal to 100. So really, this is
something right over here-- if I say something squared
is equal to 100, that means that that something is
one of the square roots of 100. And we know that 100
has two square roots. It has positive 10 and
it has negative 10. So we could say that x
plus 5, the something that we were
squaring, that must be one of the square roots of 100. So that must be equal to the
plus or minus square root of 100, or plus or minus 10. Or we could separate it out. We could say that x
plus 5 is equal to 10, or x plus 5 is equal
to negative 10. On this side right
here, I can just subtract 5 from both
sides of this equation and I would get-- I'll
just write it out. Subtracting 5 from both
sides, I get x is equal to 5. And over here, I
could subtract 5 from both sides again-- I
subtracted 5 in both cases-- subtract 5 again and I can
get x is equal to negative 15. So those are my two
solutions that I got to solve this equation. We can verify that they actually
work, and I'll do that in blue. So let's try with 5. I'll just do one of them. I'll leave the
other one for you. I'll leave the other one for
you to verify that it works. So 4 times x squared. So 4 times 25 plus
40 times 5 minus 300 needs to be equal to 0. 4 times 25 is 100. 40 times 5 is 200. We're going to
subtract that 300. 100 plus 200 minus 300,
that definitely equals 0. So x equals 5 worked. And I think you'll find that
x equals negative 15 will also work when you substitute it
into this right over here.