The Tricksters: Squares

Algebra Level 1

True or False?

x 2 = ( − x ) 2 ∀ x ∈ R \large x^2=(-x)^2 \ \ \forall x \in \mathbb{R}

Inspiration

Sometimes true True False

Nihar Mahajan by Nihar Mahajan , 20, India

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3 solutions

Nihar Mahajan
Aug 28, 2015

x 2 = ( − x ) 2 ⇒ x 2 = ( − x ) 2 ⇒ ∣ x ∣ = ∣ − x ∣ ⇒ d e f i n i t e l y t r u e . x^2=(-x)^2 \\ \Rightarrow \sqrt{x^2}=\sqrt{(-x)^2} \\ \Rightarrow |x|=|-x| \Rightarrow \ definitely \ true.

Thus the given equation is true for all reals.

7 Helpful 0 Interesting 0 Brilliant 0 Confused

yes, reals...

Hjalmar Orellana Soto - 5 years, 9 months ago
Rawan Medhat
Aug 29, 2015

( − x 2 ) (-x^2) = − 1 2 -1^2 x x 2 x^2 = x 2 x^2

hence the statement is true

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ravi Dwivedi
Aug 29, 2015

x 2 = ( − x ) 2 x^2=(-x)^2

⟹ x 2 − ( − x ) 2 = 0 \implies x^2-(-x)^2=0

Using a 2 − b 2 = ( a − b ) ( a + b ) a^2-b^2=(a-b)(a+b)

⟹ ( 2 x ) ( 0 ) = 0 \implies (2x)(0)=0 which is true

Hence the given equation is true for all reals.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

You started off with that as your assumption (call it A), and then reached an implication (call it B), and then reached another implication (call it C).

You have A → B \righarrow C A \rightarrow B \righarrow C , and while C is true, that tells us nothing about A A .

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