State whether the following equations are sometimes, always or never true.
a) a^2-2ab-b^2= (a-b)^2, bis not equal to 0
b) a^2+ b^2= (a+b)(a+b)
c) a^2-b^2= a^2-2ab+b^2
d) (a=b)^2 = a^2+2ab+b^2
I don't understand what these questions are exactly asking. Could someone please explain?
it is asking if the left is the same as the right
look at a
(a-b)(a-b) = a^2 -2ab + b^2 Always
then b
(a+b)(a+b) = a^2 + 2ab + b^2 NOT a^2+b^2
and is ONLY true if a or b is zero
same problem with c
You have a typo in d but if that is a + sign then true (a+b)(a+b)=a^2+2ab+b^2
Thank you
Certainly! These questions are asking you to determine whether each given equation is true in every case (always true), true in some cases but not all (sometimes true), or never true. To answer these questions, we can simplify each equation and analyze the resulting expression.
a) a^2-2ab-b^2= (a-b)^2, bis not equal to 0
To check if this equation is always true, we can simplify both sides of the equation and see if they are equivalent.
a^2-2ab-b^2 can be simplified by using the identity (a-b)^2 = a^2 - 2ab + b^2.
So, we rewrite the left side of the equation as (a-b)^2:
(a-b)^2 = (a-b)^2,
Now, we know that a^2-2ab-b^2 is equivalent to (a-b)^2, and the given condition specifies that b is not equal to 0. Therefore, this equation is always true as long as b is not equal to 0.
b) a^2+ b^2 = (a+b)(a+b)
To check if this equation is always true, we can simplify both sides of the equation and see if they are equivalent.
Expanding the right side of the equation, we have:
(a+b)(a+b) = a^2 + 2ab + b^2.
Now, we see that this equation is not equivalent to a^2 + b^2. Therefore, the equation is never true for all values of a and b. It may be true for certain values, but not always.
c) a^2 - b^2 = a^2 - 2ab + b^2
To check if this equation is always true, we can simplify both sides of the equation and see if they are equivalent.
Expanding the right side of the equation, we have:
a^2 - 2ab + b^2.
Now, we see that a^2 - b^2 is equivalent to a^2 - 2ab + b^2. Therefore, this equation is always true for all values of a and b.
d) (a=b)^2 = a^2 + 2ab + b^2
To check if this equation is always true, we can simplify both sides of the equation and see if they are equivalent.
Expanding the right side of the equation, we have:
(a^2 + 2ab + b^2).
Now, we see that (a=b)^2 is not equivalent to a^2 + 2ab + b^2. Therefore, this equation is never true for all values of a and b.
In summary:
a) Sometimes true, as long as b is not equal to 0.
b) Never true.
c) Always true.
d) Never true.