Go ahead, shot this: questioning a high school math teacher for the solitary student error across all grade levels that"s the many infuriating. I"d bet the the most common answer you get is some variation of exactly how students erroneously square the amount of 2 numbers (as in (a + b)2) to gain the amount of the squares that the individual number (as in a2 + b2). In fact, though, those expressions are not equal; that"s not just how squaring works. Indeed, in mine experience, students who have the right to absorb the intuition about why they room not equal space much an ext likely to success in future. This lesson will try to make feeling out the this inequality:

Before explaining why it isn"t true, i think it"s essential to recognize why for this reason many people think that is true. There are several reasons a student can think it"s true, and also each of those reasons has actually its very own logic.

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(1) I know it"s true because that multiplication:

so why isn"t the true if the symbol between them is addition instead?

This is a classic case of a college student paying an ext attention come the notation 보다 to what the signs actually mean. Squaring method multiplying an expression time itself: (ab)2 means (ab)(ab). But crucially, that"s a entirety bunch the multiplications. And also we all understand that you have the right to regroup and also rearrange items being multiplied at will (formally, we say the multiplication is associative and commutative). And also when you do regroup and also rearrange, you view there room two a"s and also two b"s all multiplied together; hence a2b2. But look in ~ what (a + b)2 means: (a + b)(a + b). Where"s the possibility for rearranging therefore simply? It"s no there. The mixture that plus and also times renders this expression more difficult to occupational with than if it"s every multiplication.

(2) I recognize it"s legit to perform this thing called "distributing":

so why can"t ns distribute the exponent just like I have the right to distribute that funny symbol? (What is that thing, anyway?)

That thing is a resources Greek letter "psi", and also I offered it to show how anything deserve to be distributed, even if that looks facility (we"ll require that idea in a minute). However many student don"t know that words "distribute" is in reality a nickname because that a much longer description of the process shown above: it"s the "distributive home of multiplication end addition". Distributing leaflets over a parking lot means that every human being in the parking lot it s okay a leaflet, and distributing multiplication end addition way that every element of the enhancement (the a and also the b) it s okay multiplied. It"s simply a fact about our number device that you can choose to include two numbers prior to you main point the amount by a 3rd number (the left next of the over equation), or you deserve to multiply every of the two numbers by the 3rd number individually very first (the right side), then add the products, and you"ll get the exact same result. In fact, the equation in (1) above is a demonstrate of a different distributive property: "the distributive residential property of exponents end multiplication". And that one is true due to the fact that multiplication is associative and commutative.

But the "distributive residential or commercial property of exponents end addition" is merely not valid. Miscellaneous distributive properties are various mathematical processes (a subtlety you can miss if girlfriend don"t understand their complete names), and also they don"t all need to be valid. In fact, many are not; any one that is valid is special and also important.

(3) But...it simply seems prefer they should be equal!

Well...only at an initial maybe. It"s a common misconception that mathematical processes that look appropriate on the page or seem appropriate in her head have to be true. Mathematicians strategy it in the completely opposite direction—in a way, nothing should be true until you can prove it"s true. And also equality in this instance just...isn"t true! as well bad! the end of luck! oh well, we"ll learn to attend to it!

Once you look at this situation really carefully, making use of examples and also various different mathematical interpretations, you"ll find that you most likely don"t yes, really even believe that the two sides can be equal. You currently have the expertise to convince you yourself of that. Let"s look at at some of those examples.

If you"re in a candy store scooping candy into a bag, and also you to buy one-and-a-half pounds of liquid that costs $1.50 per pound, just how much will you acquire charged? You could not recognize the answer off the peak of your head, however I"ll gambling you understand that $1.25 is a ridiculous answer. You"re buying much more than a lb of candy, for this reason the price would simply have to be much more than $1.50. However if you think the the square that a amount is the sum of the separation, personal, instance squares, you"d have actually to think that $1.25 is right:

One-and-a-half pounds times one-and-a-half dollars per pound doesn"t equal one-and-a-quarter dollars. Note the critical placement of the not-equals-sign. Be sure you know the factor for every step of the line the math.

I"ll bet you deserve to probably square the number 20 in her head. 2 time 20 is 40, for this reason 20 time 20 is 400. You might not it is in as rapid with squaring 21. The actual prize isn"t important, yet do you have the intuition that it"s not 401? Well that intuition comes from a deep-down intuition that the square that a sum is no the sum of the individual squares:

Again, some component of your brain already knows the (20 + 1)2 isn"t the exact same as 202 + 12.

What does (a + b)2 equal, then? Or, put one more way, is over there an expression there is no parentheses that constantly has the precise same value? The officially algebra is below. It provides the distributive property of multiplication over addition that we saw above. The famous term because that the process that you could know is FOILing:

Again, be certain you understand every step in that process.

Now, a2 + 2ab + b2 isn"t exactly an obvious method to rewrite (a + b)2, but it does have actually the advantage of gift correct! If you try to interpret that in a different way, though (geometrically, because that example), it deserve to make much more sense. The natural way to know the concept of squaring is with looking at the area of a square—which is calculated by squaring. So below is a photo of a square whose sides space each a + b long. To make that more clear, those political parties are damaged up into their separate a and b parts.

Asking about (a + b)2, then, is as with asking about the area of that entirety square. However the totality square is broken up right into smaller squares and also rectangles, and we know sufficient information to calculation each of those smaller components separately. The locations of the two smaller squares room calculated below.

Notice that the areas of the two smaller squares with each other come nowhere close to totaling the area the the big square. In algebra terms, we"d need to say the (a + b)2 must merely be better than a2 + b2. Of food that way they can"t it is in equal, which is specifically what we"ve to be trying come understand! This snapshot actually speak us also more, though. It speak us exactly how much greater. Every of the blue rectangles has a length of a and a broad of b, therefore they each have an area of a times b. And there"s 2 of them. Which way precisely that (a + b)2 = a2 + 2ab + b2, simply as we experienced in the algebra.

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Finally, I"ll present one more means to recognize the original inequality. This last method requires that you know what the Pythagorean theorem says, and I"ll assume below that you do (if not, you can skip this part). Note the square constructed on the hypotenuse of the triangle below:


The square has actually an area the c2, i beg your pardon the Pythagorean theorem says is equal to a2 + b2. However that is specifically the ideal hand side of our initial inequality. Therefore it renders sense to ask about the square represented by the left hand side. A square v that area would have to have a side length of a + b. But it"s clear from looking in ~ the triangle the a + b needs to be bigger 보다 c (walking along the hypotenuse should require fewer procedures than walking along the foot of the triangle—technically that"s referred to as "the triangle inequality"). That is, the two sides of the inequality every geometrically stand for the area that a square, but those squares can"t be the very same size, for this reason the two expressions can"t be equal.