Question

This problem addresses some common algebraic errors. For the equalities stated below assume that x and y stand for real numbers. Assume that any denominators are non-zero. Mark the equalities with T (true) if they are true for all values of x and y, and F (false) otherwise.

1. (x+y)^2 =x^2+y^2 __

2. (x+y)^2 = x^2 +2xy+y^2__

3. x/x+y=1/y__

4. x−(x+y) = y__

5. √x^2 =x__

6. √x^2 = |x|__

7. √x^2+4=x+2__

8. 1/x+y=1/x+1/y__

Answer 1

1. F

observe that
`(5+2)^2=49 \\eq 29=5^2+2^2`

2. T

Let x and y real numbers.

`(x+y)^2=(x+y)(x+y)=x^2+2xy+y^2`

3. F

Observe that if x=3 and y=2
`(3)/(3+2)=(3)/(5)\\eq (1)/(2)`

4. F

If x=y=3,
`3-(3+3)=3-6=-3\\eq 3`

5. F

if x=-1,
`√(-1^2)=√(1)=1\\eq -1`

6. T

7. F

if x=-1,
`√(-1^2+4)?√(5)\\eq 1=-1+2`

8. F

If x=1 and y=2,
`(1)/(1+2)=(1)/(3)\\eq (3)/(2)=(1)/(1)+(1)/(2)`