In Exercises 83–90, perform the indicated operation or operations...

Question

In Exercises 83–90, perform the indicated operation or operations. (3x+4y)^2−(3x−4y)^2

Accepted Answer

Hello today we're going to be simplifying an algebraic expression. So the algebraic expression given to us is the quantity five X plus y squared minus the quantity five X minus y squared. Now, in order to start this we're going to need to expand five X plus Y squared and five x minus y squared. So let's go ahead and do that. So five X plus y squared means that we have five X plus Y times itself two times. So we can rewrite this as five X plus Y times five X plus Y. Now, in order to expand this, we're going to need to foil the quantities together. So we're going to take our first term five X and multiply to the 1st and 2nd term of our second quantity and we're taking our second term Y and multiplying it to the 1st and 2nd term of our second quantity. Doing so five X times five X is going to give me 25 X squared five X times Y is going to give me five X y y times five X. Is going to give me five X Y and Y times Y is going to give me Y squared. Now what we need to do is go ahead and combine like terms, the only combine like terms here is five X Y and five X Y. Because they both have the xy term and combining this together is going to give us the try no meal 25 X squared plus 10 X Y plus Y squared. So we've expanded the left hand side of this algebraic expression. Now let's go ahead and expand the right hand side. So the right hand side is five x minus Y squared. And just like five X plus Y squared. This means that we have the quantity five x minus Y times itself two times. So this could be rewritten as five x minus Y times five x minus y. And once again we're going to need to follow these quantities together in order to expand them. So we're gonna take five X. Multiply to our 1st and 2nd term here and we're taking negative Y and multiplying it to our 1st and 2nd term. Doing that is going to give us five X times five X which is 25 X squared five X times negative Y which is negative five X Y negative Y times five X. Which is going to give me negative five X Y. And negative Y times negative Y. Which is going to give me positive Y squared. Now, the only combine herbal light terms here is negative five X Y. And negative five X. Y. Because of the xy term. And combining those together is going to give me the tri no meal 25 X squared minus 10 X Y plus Y squared. And that's gonna be the right hand side of this algebraic expression simplified. All right. Now keep in mind that these two quantities are being subtracted from each other. So we're taking our left hand side of this algebraic expression and subtracting the right hand side from it. Rewriting that is going to give me 25 X squared plus 10 X Y plus y squared minus the quantity. 25 x squared minus 10 X Y plus y squared. Now, in order to open up these parentheses, I'm going to need to take our negative sign and distributed to every term in this. Try no meal here. Doing so is going to give me 25 X squared plus 10 X Y plus Y squared minus 25 X squared plus 10 X Y. Because a negative times a negative is a positive number and minus Y squared because a negative times a positive is going to produce a negative number. All right now we need to go ahead and combine the like terms to simplify this. Well, 25 X squared and a negative 25 X squared are just gonna cancel each other out. And positive Y squared and negative Y squared are going to cancel each other out. So what we are left with is 10 X Y plus 10 X Y and 10 X Y plus 10 X Y is going to give me 20 X Y. And that's going to be the simplified version of this algebraic expression. And that also means that the answer to this problem is going to be C. So I hope this video helps you in understanding how to simplify an algebraic expression and I'll go ahead and see you all in the next video.

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