```The √ symbol means radical (or square root)
(a) x2 _ 2x - 13= 0
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x2 _ 2x
=13
multiply by 4
2_
4 x 8x =52
4 x2 _ 8x +4=56
take square root
2x-2 = ± 2√14
divide by 2
x-1 = ± √14
solve for x
x = 1 + √14,
x = 1 - √14.
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(b) 4 x2 - 4x + 3 =0
4 x2 - 4x = -3
multiply by 16
64x2 - 64x = --48
2
64x - 64x+16 = -32
take square root
8x-4= ± 4i√2
divide by 4
2x-1 = ± i√2
solve for x
x = (1- i√2)/2, x = (1+i√2)/2
In this case the roots are complex nonreal.
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c) x2 + 12x - 64= 0
x2 + 12x = 64
4x2 + 48x = 256
4x2 + 48x +144 = 400
2x+12 = ± 20
x+6 = ± 10
x= - 16, x = 4.
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(d) 2x2 -_ 3x - 5 =0
2x2 _ 3x =5
16x2 _ 24x =40
16x2 _ 24x+9 =49
4x-3 = ±7
4x =3+7, 4x= 3-7
x = 5/2, x= - 1.
multiply by 4
take square root
divide by 2
solve for x
multiply by 8
take square root
solve for x
2. Mathematicians have been searching for a formula that yields prime numbers. One such
formula was x2 _ x + 41. Select some numbers for x, substitute them in the formula, and see if the
prime numbers occur. Try to find a number for x that when substituted in the formula yields a
composite number.
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
x2+x+41
41
43
47
53
61
71
83
97
113
131
151
173
197
223
251
281
313
347
383
421
461
503
547
593
Prime?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
The first 24 integer values of x give prime
values of x2+x+41 as it can be seen from the table.
However it can be seen that this is not always the
case. For instance, write
x2+x+41=x (x+1)+41.
If one chooses x such that either x or (x+1) is a
multiple of 41 then x2+x+41 is not a prime number
anymore.
For instance when x=40 we have x2+x+41=412, not
a prime number anymore.
Similarly, when x=41 we have x2+x+41=41*43
not a prime number anymore.
Other values of x for which x2+x+41is not prime
are x= 81, 82, 122, 123, ….etc.
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