What is Completing the Square?

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Answer 1 · 2018-04-05T13:53:42

See explanation below

When you have a polynomial such as

$x^{2} + 4 x + 20$ $x^2+4x+20$

it is sometimes desirable to express it in the form of

$a^{2} + b^{2}$ $a^2 + b^2$

To do this, we can artificially introduce a constant which allows us to factor a perfect square out of the expression like so:

$x^{2} + 4 x + 20$ $x^2+4x+20$

$= x^{2} + 4 x + 4 - 4 + 20$ $=x^2+4x+color(red)4-color(green)4+20$

Notice that by simultaneously adding and subtracting $4$ $4$ , we have not changed the value of the expression.

Now we can do this:

$= (x^{2} + 4 x + 4) + (20 - 4)$ $=(x^2+4x+color(red)4)+(20-color(green)4)$

$= {(x + 2)}^{2} + 16$ $=(x+2)^2+16$

$= {(x + 2)}^{2} + 4^{2}$ $=(x+2)^2+4^2$

We have "completed the square"!