Squaring both members of an inequality is often a useful technique to eliminate square roots, but it possibly reverses the sense of an inequality if one member is negative, and certainly reverses it if both members are negative. To get the right answer, the problem must be broken into numerous cases according to the signs of members.
Left Member
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The question here is to determine Θ2 in:
q √(10 + √20) + s √(10 − √20) Θ2 0 [#2]
Observe that 10 + √20 and 10 − √20 are themselves positive, and so are their square roots.
Θ2 in #2
| s < 0
| s = 0
| s > 0
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q < 0
| Θ2 is less-than | Θ2 is less-than | see case alpha
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q = 0
| Θ2 is less-than | Θ2 is equal-to | Θ2 is greater-than
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q > 0
| see case beta | Θ2 is greater-than | Θ2 is greater-than
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Case Alpha |
Equivalent to #2 is:
s √(10 − √20) Θ2 −q √(10 + √20)
Both members are positive, so square it without exchange:
s2(10 − √20) Θ2 q2(10 + √20)
Rearrange:
10(s2 − q2) Θ2 √20(s2 + q2) [#3]
Θ2 in #3
| s2 + q2 = 0
| s2 + q2 > 0
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s2
− q2 < 0
| Θ2 is less-than | Θ2 is less-than
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s2
− q2 = 0
| Θ2 is equal-to | Θ2 is less-than
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s2
− q2 > 0
| Θ2 is greater-than | see below
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When in #3 both members are positive, square it without exchange:
100(s2 − q2)2 Θ2 20(s2 + q2)2
which is all integers, and thus can easily be evaluated exactly.
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Case Beta |
Recall:
s √(10 − √20) Θ2 −q √(10 + √20)
Both members are negative, so square it with exchange:
q2(10 + √20) Θ2 s2(10 − √20)
Rearrange:
√20(s2 + q2) Θ2 10(s2 − q2) [#4]
Θ2 in #4
| s2 − q2 < 0
| s2 − q2 = 0
| s2 − q2 > 0
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s2
+ q2 = 0
| Θ2 is greater-than | Θ2 is equal-to | Θ2 is less-than
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s2
+ q2 > 0
| Θ2 is greater-than | Θ2 is greater-than | see below
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When in #4 both members are positive, square it without exchange:
20(s2 + q2)2 Θ2 100(s2 − q2)2
which is all integers.
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Right Member |
The question here is to determine Θ3 in:
− r (√5 + 1) − t (√5 − 1) Θ3 0
or equivalently
(t − r) Θ3 √5(t + r) [#5]
Θ3 in #5
| t + r < 0
| t + r = 0
| t + r > 0
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t − r < 0
| see case gamma
| Θ3 is less-than
| Θ3 is less-than
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t − r = 0
| Θ3 is greater-than
| Θ3 is equal-to
| Θ3 is less-than
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t − r > 0
| Θ3 is greater-than
| Θ3 is greater-than
| see case delta
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Case Gamma |
In #5 both members are negative, so square it with exchange:
5(t + r)2 Θ3 (t − r)2
which is all integers.
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Case Delta |
In #5 both members are positive, so square it without exchange:
(t − r)2 Θ3 5(t + r)2
which is all integers.
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Summary |
With Θ2 and Θ3 established, resolution of Θ1 can begin. This table handles many cases:
Θ1 in #1
| Θ3 is less-than
| Θ3 is equal-to
| Θ3 is greater-than
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Θ2 is less-than
| see case epsilon | Θ1 is less-than | Θ1 is less-than
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Θ2 is equal-to
| Θ1 is greater-than | Θ1 is equal-to | Θ1 is less-than
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Θ2 is greater-than
| Θ1 is greater-than | Θ1 is greater-than | see case zeta
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Case Epsilon |
In #1 both members are negative, so square it with exchange:
r2 (6 + 2 √5) + t2 (6 − 2 √5) + 8 r t Θ1 q2 (10 + √20) + s2 (10 − √20) + 2 q s √80
Move the radicals left, and divide by two:
√5(r2 − t2 − q2 + s2 − 4 q s) Θ1 − 3 r2 − 4 r t + 5 q2 − 3 t2 + 5 s2
Define
X = r2 − t2 − q2 + s2 − 4 q s
and
Y = − 3 r2 − 4 r t + 5 q2 − 3 t2 + 5 s2
This consolidates the problem to
√5X Θ1 Y [#6]
Θ1 in #6
| Y < 0
| Y = 0
| Y > 0
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X < 0
| same as Y2 Θ1 5X2 | Θ1 is less-than | Θ1 is less-than
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X = 0
| Θ1 is greater-than | Θ1 is equal-to | Θ1 is less-than
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X > 0
| Θ1 is greater-than | Θ1 is greater-than | same as 5X2 Θ1 Y2
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Case Zeta |
In #1 both members are positive, so square it without exchanging:
q2 (10 + √20) + s2 (10 − √20) + 2 q s √80
Θ1
r2 (6 + 2 √5) + t2 (6 − 2 √5) + 8 r t
Move the radicals right, and divide by two:
− 3 r2 − 4 r t + 5 q2 − 3 t2 + 5 s2 Θ1 √5(r2 − t2 − q2 + s2 − 4 q s)
Again define
X = r2 − t2 − q2 + s2 − 4 q s
and
Y = − 3 r2 − 4 r t + 5 q2 − 3 t2 + 5 s2
This consolidates the problem toY Θ1 √5X [#7]
Θ1 in #7
| X < 0
| X = 0
| X > 0
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Y < 0
| same as 5X2 Θ1 Y2 | Θ1 is less-than | Θ1 is less-than
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Y = 0
| Θ1 is greater-than | Θ1 is equal-to | Θ1 is less-than
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Y > 0
| Θ1 is greater-than | Θ1 is greater-than | same as Y2 Θ1 5X2
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