+35 Let's take a look at \((x + y)^2\) and \(x^2 + y^2\) While Beeker believes that these two expressions are equal for all real numbers x and y Clod believes they are not! Let's get to the bottom of this!
a) Evaluate \((x + y)^2\) and \(x^2 + y^2\) for x = 7 and y = 11
b) For which values of x and y does \((x + y)^2\) equal \(x^2 + y^2\) ? For which values of x and y does \((x + y)^2\) not equal \(x^2 + y^2\)?
I solved question a , the first expression equal to 324 and the second one equal to 170.
For question b, I set two equation equal to each other first, and then I get 2xy=0.
I get this two expression equal when x=0, y=0 and x=-y, y=-x.
I also get this two expressions does not equal when x and y are both non-zero number, and x and y are not negations of each other.
I wanna ask is this statement (one of the x and y equal to zero, and the other one can equal to all teal #s) correct in the situiation of this two expressions are equal. \(\)\(\)
a) Let's evaluate 2x^3 + 5y^2 and x^3 + 5xy^2 for x = 7 and y = 11.
For x = 7, ad y = 11:
2(7)^3 + 5(11)^2 = 2(343) + 5(121) = 686 + 605 = 1291
for the second expression:
(7)^3 + 5(7)(11)^2 = 343 + 5(7)(121) = 343 + 5(847) = 343 + 4235 = 4578
So the first expresion is equal to 1291 and the second one is equal to 4578.
b) Now, let's find the values of x and y for which 2x^3 + 5y^2 is equal to x^3 + 5xy^2 and also the values for which they are not equal.
To find the values for which they are equal, we set the two expressions equal to each other:
2x^3 + 5y^2 = x^3 + 5xy^2
Simplifying the equatoin:
2x^3 - x^3 = 5y^2 - 5y^2
x^3 = 5y^2(x - 1)
Now, we can see two scenes where the expressions are equal:
When x = 0, the equaton becomes 0 = 0, which is true for all y (all real numbers).
When x = -y, the equation becomes (-y)^3 = 5y^2(-y - 1).
Simplifying further:
-y^3 = -5y^3 - 5y^2
4y^3 + 5y^2 = 0
This equation is true when y = 0 or y = -5/4.
So, the values for which the expressions are equal are: (x = 0, y is any real number) and (x = -y, y = 0 or y = -5/4).
On the other hand the values for which the expressoins are not equal, we will consider any non-zero value of x and y that do not satisfy the condtions we found above So if x and y are both nonzero numbers and they are not negotiations of each other (x is not eqaul to -y) then the expressions are equal.
