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Worked example: Rewriting expressions by completing the square

Sal rewrites x²+16x+9 as (x+8)²-55 by completing the square.

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• At , why did you subtract 64 from the other side, where in other videos you added 64? What you do to one side you do to the other?
• Sal doesn't have an equation, so there is no other side. He is working with an expression. To ensure he maintains the original value of the expression, he can only add a value that equal 0 (this is the identify property of addition). This is done by adding the 64 and then also subtracting the 64. 64 - 64 = 0. Thus, he hasn't the expression. He's just making it look different.
Hope this helps.
• what does the b mean? I know it is a constant but, in the equation, what is it?
• This is what is known as the vertex form of a quadratic equation: it is for finding the vertex (obviously).
Given the expression (x+a)^2+b, the vertex of this quadratic is (-a,b). So, the b-value is the y-value of the vertex of the quadratic.
For example:
(x-7)^2 - 23
a = -7 and b = -23
So, the vertex of this quadratic is (-(-7),-23), or (7,-23).
• At , how did he geat a=8?
• He tried to find what(a) times 2 = 16x.
So you could simple multiplication to get 8*2=16.
you could also do 16/2=8 (easier for bigger numbers).
Hope this helps :)
• At about , Sal adds 64, and subtracts 64. There is no equal sign in the expression, so what he's basically doing is making a zero pair?? Uhhh, why would he do that, again?? And then he factors out the (x+a)² part (that I understand), but when he subtracts the 64 from 9... umm, what's the point of doing that? And isn't the WHOLE THING like, an expression?? Huh. So...why does he have to worry about making the expression true (when he added the 64, he said that he had to subtract a 64) when there's no equal sign?? Very confusing.
• Even though there is no equal sign we are trying to solve this particular expression, or rather find the zeroes of it. Since we want to do it for this particular expression we want to make sure it doesn't change in value, so basically if you graphed x² + 16x + 9, it would be the same graph as x² + 16x + 64 + 9 - 64. Now since these are the same graphs anything we do to the second expression that keeps its value the same will ake it true for the original too.

Now for that second one if we focus on x² + 16x + 64 hopefully you can see why it turns into (x+8)². Then that just leaves +9 - 64. You could leave that as is and still be right, but there's no reason not to combine like terms.

Basically if you expanded (x + 8)² - 55 you would get back to x² + 16x + 9, which is the whole point of this. This completeing the square method is just a method to change from one form of quadratics to another.

Let me kno if something did not make sense though.
• In the last video it was (x-a)^2 and in this video it is (x+a)^2. How do you decide which one to use?
• Well, it depends on your quadratic equation. (x+a)² = x²+ 2ax + a² like in this video. The previous video's equation was of the form x² - 2ax + a², which is the same as (x - a)².
• it's simple, but why choose that formula?
• Completing the Square is quite useful for those equations that are not simply factorable. Also, the quadratic formula is mainly only good for expressions where a>1.
• Is completing the square just writing a quadratic in vertex form?
(1 vote)
• Completing the square can be used for multiple things, here are a few:
1) Solving a quadratic equation that has 1 variable.
2) Converting a quadratic function / equation that has 2 variables into vertex form.
3) Changing the equation of a circle into standard form.
• Sal has saved me again
• Is this supposed to be a way to rewrite quadratic equations into vertex form when I'm trying to graph a parabola?
• Sal calls this technique "Completing the square".

Is this form called "Perfect Square" form?:
(x+a)^2+b

Also,

What do we call this form?:
x^2 +2ax +a^2 +b

Is it just called "Completing the square" form?

I'm trying to memorize these forms so I can recall them easier but not sure what they're named.

Thank you K.A. and learners
(1 vote)
• y=a(x-h)^2+k (similar to your "perfect square" form is actually called vertex form where a is a scale factor and (h,k) is the vertex. Your example just has a=1 and different labels for the vertex which would be at (-a,b). The other two forms are standard y=ax^2+bx+c and factored form y=(ax+b)(cx+d). These are the three forms of quadratics. The idea of a^2x^2 + 2abx + b^2 is just a special case of the standard form which happens to be simple to get into factored form (ax+b)^2. Same with difference of perfect squares where a^2 x^2 - b^2 factors to (ax+b)(ax-b). There are only three forms, there is no perfect square form or completing the square form.