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worty [1.4K]
3 years ago
7

Write x^2 − 2x − 3 = 0 in the form (x − a)^2 = b, where a and b are integers.

Mathematics
1 answer:
Airida [17]3 years ago
5 0
X²-2x-3 = 0     let's start by grouping the "x"s

(x² - 2x + [?])² - 3 = 0

so, we have a missing fellow there, in order to make the group, a perfect square trinomial, namely to "complete the square", hmmm so the tell-tale fellow is the middle term.

from a perfect square trinomial we know that the middle term is a product of 2 and the "term on the left" and the "term on the right", like 

\bf \begin{array}{cccccllllll}
 a^2& + &2 a b&+& b^2\\
\downarrow && &&\downarrow \\
 a&& && b\\
&\to &( a +  b)^2&\leftarrow 
\end{array}\qquad \qquad \quad 
\begin{array}{cccccllllll}
 a^2& - &2 a b&+& b^2\\
\downarrow && &&\downarrow \\
 a&& && b\\
&\to &( a -  b)^2&\leftarrow 
\end{array}

\bf \textit{therefore}\qquad 2(x)(\boxed{?})=\stackrel{middle~term}{2x}\implies \boxed{?}=\cfrac{2x}{2x}\implies \boxed{?}=1

aha!!  so our missing fellow is 1.

now, let's keep in mind that all we're doing is borrowing from our very good friend Mr Zero, 0.  So if we add 1², we also have to subtract 1².

(x² - 2x + 1² - 1²) - 3 = 0

(x² - 2x + 1) -1 -3 =0

(x - 1)² - 4 = 0

(x - 1)² = 4
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Answer:

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Step-by-step explanation:

A) The sequence is given as;

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The differences are:

5,5,5,5.

This is an arithmetic sequence following the formula;

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d is 5

Thus;

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Now, a_1 = 7. Thus;

a_n = 7 + 5n - 5

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B) The sequence is given as;

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Now, let's write out the differences of this sequence:

Differences are:

4, 8, 16, 32

This shows that it is a geometric sequence with a common ratio of 2.

In the given sequence, a_1 = 3 and a_2 = 7 and a_3 = 15

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Also, a_(2 + 1) = 2a_2 + 1

Combining both equations, we can deduce that: a_(n + 1) = 2a_n + 1

Thus; a_n can be expressed as:

a_n = (2^(n + 1)) - 1

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3 years ago
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