SIMPLIFYING $\,(a+b)^2\,$ and $\,(a-b)^2$

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Here are two very important and common expressions: [beautiful math coming... please be patient] $$ \overset{\text{in one step, be able to go from here ...}}{\overbrace{(a+b)^2}} = (a+b)(a+b) = a^2 + ab + ab + b^2 = \ \ \ \ \ \ \ \ \overset{\text{... to here}}{\overbrace{a^2 + 2ab + b^2}} $$ [beautiful math coming... please be patient] $$ \overset{\text{in one step, be able to go from here ...}}{\overbrace{(a-b)^2}} = (a-b)(a-b) = a^2 - ab - ab + b^2 = \ \ \ \ \ \ \ \ \overset{\text{...to here}}{\overbrace{a^2 - 2ab + b^2}} $$

You should (eventually) be able to multiply out expressions like these without writing down any intermediate results.

Be careful!

One of the most common algebra mistakes is to think that
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$\,(a + b)^2\,$ is equal to
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$\,a^2 + b^2\,$.

NOT SO!!!

You've got the Firsts (‘F’) and the Lasts (‘L’)

but have left out the Outers (‘O’) and the Inners (‘I’)!

One of my students (Ian Sullivan) came up with this memory device

as a reminder that the squares
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$\,(a+b)^2\,$ and $\,(a-b)^2\,$ require **FOIL**ing:

EXAMPLES:

Simplify:
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$(x-2)^2$

Answer:
x^2 - 4x + 4

Note: Use the ‘ ^ ’ key for exponents.

Input your answer in the most conventional way.

That is, even though 4 - 4x + x^2 is a correct answer,

it isn't recognized as correct by this program.

Note: Use the ‘ ^ ’ key for exponents.

Input your answer in the most conventional way.

That is, even though 4 - 4x + x^2 is a correct answer,

it isn't recognized as correct by this program.

Simplify:
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$(3x+y)^2$

Answer:
9x^2 + 6xy + y^2

Note: Variables*must* be typed in the order they appear,

going from left to right, for your answer to be recognized as correct.

That is, even though 9x^2 + 6yx + y^2 is a correct answer,

it isn't recognized as correct by this program.

Note: Variables

going from left to right, for your answer to be recognized as correct.

That is, even though 9x^2 + 6yx + y^2 is a correct answer,

it isn't recognized as correct by this program.

Master the ideas from this section

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Simplifying Expressions like $\,(a - b)(c + d - e)$

by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:

Simplifying Expressions like $\,(a - b)(c + d - e)$