Add all the Equations?
⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ a 2 + 2 b = 7 b 2 + 4 c = − 7 c 2 + 6 a = − 1 4
a , b and c are real numbers that satisfy the system of equations above. What is the value of a 2 + b 2 + c 2 ?
The answer is 14.
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2 solutions
i did the same thing!!
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Yes, I noticed. I just wanted to show it more clearly.
Add all the equations as the topic suggests.We get,
a
2
+
b
2
+
c
2
+
6
a
+
2
b
+
4
c
+
1
4
=
0
Split the
1
4
as
1
+
4
+
9
.
(
a
2
+
6
a
+
9
)
+
(
b
2
+
2
b
+
1
)
+
(
c
2
+
4
c
+
4
)
=
0
⇒
(
a
+
3
)
2
+
(
b
+
1
)
2
+
(
c
+
2
)
2
=
0
.
Since the sum of the squares are equal to
0
,
each term should also equal to
0
.
(
a
+
3
)
2
=
0
⇒
a
=
−
3
.
(
b
+
1
)
2
=
0
⇒
b
=
−
1
.
(
c
+
2
)
2
=
0
⇒
c
=
−
2
.
Therefore,
(
a
2
+
b
2
+
c
2
)
=
1
4
.
You need to mention that a , b , c are real. If not there are many complex solutions such that square of them is negative.
⎩ ⎪ ⎨ ⎪ ⎧ a 2 + 2 b = 7 b 2 + 4 c = − 7 c 2 + 6 a = − 1 4 . . . ( 1 ) . . . ( 2 ) . . . ( 3 )
( 1 ) + ( 2 ) + ( 3 ) : a 2 + 6 a + b 2 + 2 b + c 2 + 4 c a 2 + 6 a + 9 + b 2 + 2 b + 1 + c 2 + 4 c + 4 − 9 − 1 − 4 ( a + 3 ) 2 + ( b + 1 ) 2 + ( c + 2 ) 2 − 1 4 ( a + 3 ) 2 + ( b + 1 ) 2 + ( c + 2 ) 2 = − 1 4 = − 1 4 = − 1 4 = 0
We note that the LHS are squared terms ( a + 3 ) 2 + ( b + 1 ) 2 + ( c + 2 ) 2 ≥ 0 and equality only happens when a = − 3 , b = − 1 and c = − 2 . Therefore, a 2 + b 2 + c 2 = 9 + 1 + 4 = 1 4 .