Question

Disprove the following statements using counterexample. 

1) (𝑥 + 4)2 = 𝑥2 + 16  

2) 𝑥2 > x  

3) √𝑥2 = 𝑥  

Answer 1

1) Let $x=1$

$$(x+4)^2=(1+4)^2=25$$

$$x^2+16=1^2+16=17$$

$$25\not=17$$

Hence $(x+4)^2\not=x^2+16$ for $x=1$

2) Let $x=\dfrac{1}{2}$

$$x^2=(\dfrac{1}{2})^2=\dfrac{1}{4}<\dfrac{1}{2}=x$$

Hence $x^2<x$ for $x=\dfrac{1}{2}$

3) Let $x=-1$

$$\sqrt{x^2}=\sqrt{(-1)^2}=\sqrt{1}=1\not=-1=x$$

Hence $\sqrt{x^2}\not=x$ for $x=-1.$