# Square of Sum and Difference

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Square of sum and difference:

huge color(purple)((a+-b)^2=a^2+-2ab+b^2)

Let's see how to derive it.

Recall, that exponent is just repeating multiplication.

Thus, we can write that (a+b)^2=(a+b)(a+b).

Now, apply FOIL: (a+b)(a+b)=a*a+a*b+b*a+b*b=a^2+2ab+b^2.

Similarly, it can be shown, that (a-b)^2=a^2-2ab+b^2.

Or, more shortly: (a+-b)^2=a^2+-2ab+b^2.

Geometrically (a+b)^2 represents an area of the square with side a+b.

But, as shown on picture, this square consist of four smaller squares with areas a^2, ab, ab, b^2.

Thus, (a+b)^2=a^2+ab+ab+b^2=a^2+2ab+b^2.

Example 1. Multiply (2x+3y)^2.

Here a=2x and b=3y.

Just use above formula: (2x+3y)^2=(2x)^2+2*(2x)*(3y)+(3y)^2=4x^2+12xy+9y^2.

Let's see how to handle minus sign.

Example 2. Multiply (8/3ab-3cd)^2.

Here a=8/3ab and b=3cd.

Now, use formula for difference: (8/3ab-3cd)^2=(8/3ab)^2-2*(8/3ab)*(3cd)+(3cd)^2=64/9a^2b^2-16abcd+9c^2d^2.

Finally, let's do a slightly harder example.

Example 3. Multiply the following: (-xyz-5x^2)^2.

Till now, we didn't see two minus signs, but this case can be handled easily.

There are two options:

• a=-xyz and b=-5x^2; apply sum formula.
• a=-xyz and b=5x^2; apply difference formula.

I choose second option: (-xyz-5x^2)^2=(-xyz)^2-2*(-xyz)*(5x^2)+(5x^2)^2=x^2y^2z^2+10x^3yz+25x^4.

From last example we see, that color(purple)((-a-b)^2=(a+b)^2).

Another nice application of square of sum formula is to calculate square of a number. In many cases you can perform calculations mentally without calculator (or pen and paper).

Example 4. Calculate 24^2.

We could use calculator or multiply vertically, but there is simpler way.

We know, that 20^2=400.

Thus, 24^2=(20+4)^2=20^2+2*20*4+4^2=400+160+16=576.

Alternatively 24^2=(30-6)^2=30^2-2*30*6+6^2=900-360+36=576.

Note, that this method is not always the simplest.

Now, it is time to exercise.

Exercise 1. Multiply (5z+3y)^2.

Answer: 25z^2+30zy+9y^2.

Exercise 2. Multiply (-1/3xy^2+2x)^2.

Answer: 1/9x^2y^4-4/3x^2y^2+4x^2.

Hint: either swap summands ((-1/3xy^2+2x)^2=(2x-1/3xy^2)^2: commutative property of addition) or proceed as always.

Exercise 3. Multiply the following: (-3x-2)^2.

Answer: 9x^2+12x+4.

Exercise 4. Calculate 31^2 using square of sum/difference formula.

Answer: 961. Hint: 31^2=(30+1)^2 or 31^2=(40-9)^2 (however, first option is easier).