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yarga [219]
3 years ago
15

Which expression is equivalent to the area of metal sheet required to make this square-shaped traffic sign?

Mathematics
2 answers:
olga55 [171]3 years ago
7 0

Answer:

Hence, the expression for the area of the metal sheet used is:

(x+1)^2=x^2+2x+1

Step-by-step explanation:

It is given that a traffic signal is in the shape of a square such that its's one side is labeled as:

x+1.

We are asked to find the expression that is equivalent to the area of metal sheet required to make this square-shaped traffic sign.

The area of metal sheet used will be equal to the area of the traffic signal which is equal to the area of the square with side length (x+1).

Hence, the area of square(A) is:

A=(x+1)^2\\\\A=x^2+1+2x

Hence, the expression equivalent to the area of metal sheet used is:

(x+1)^2=x^2+2x+1

Lena [83]3 years ago
6 0
(x+1)^2
(x+1)(x+1)
x^2+x+x+1
The expression is x^2+2x+1
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This statement can be proven by contradiction for n \in \mathbb{N} (including the case where n = 0.)

\text{Let $n \in \mathbb{N}$ be a perfect square}.

\textbf{Case 1.} ~ \text{n = 0}:

\text{$n + 2 = 2$, which isn't a perfect square}.

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\textbf{Case 2.} ~ \text{$n \in \mathbb{N}$ and $n \ne 0$. Hence $n \ge 1$}.

\text{Assume that $n$ is a perfect square}.

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Step-by-step explanation:

Assume that the natural number n \in \mathbb{N} is a perfect square. Then, (by the definition of perfect squares) there should exist a natural number a (a \in \mathbb{N}) such that a^2 = n.

Assume by contradiction that n + 2 is indeed a perfect square. Then there should exist another natural number b \in \mathbb{N} such that b^2 = (n + 2).

Note, that since (n + 2) > n \ge 0, \sqrt{n + 2} > \sqrt{n}. Since b = \sqrt{n + 2} while a = \sqrt{n}, one can conclude that b > a.

Keep in mind that both a and b are natural numbers. The minimum separation between two natural numbers is 1. In other words, if b > a, then it must be true that b \ge a + 1.

Take the square of both sides, and the inequality should still be true. (To do so, start by multiplying both sides by (a + 1) and use the fact that b \ge a + 1 to make the left-hand side b^2.)

b^2 \ge (a + 1)^2.

Expand the right-hand side using the binomial theorem:

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b^2 \ge a^2 + 2\,a + 1.

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