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

Answer:

1. F

observe that $(5+2)^2=49 \neq 29=5^2+2^2$

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 $\frac{3}{3+2}=\frac{3}{5}\neq \frac{1}{2}$

4. F

If x=y=3, $3-(3+3)=3-6=-3\neq 3$

5. F

if x=-1, $\sqrt{-1^2}=\sqrt{1}=1\neq -1$

6. T

7. F

if x=-1, $\sqrt{-1^2+4}?\sqrt{5}\neq 1=-1+2$

8. F

If x=1 and y=2, $\frac{1}{1+2}=\frac{1}{3}\neq \frac{3}{2}=\frac{1}{1}+\frac{1}{2}$