a plus b plus c WHOLE SQUARE FORMULA
In this section, we are going to see the formula or expansion for (a + b + c)2.
That is,
(a + b + c)2 = (a + b + c)(a + b + c)
(a + b + c)2 = a2 + ab + ac + ab + b2 + bc + ac + bc + c2
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
Example 1 :
Expand :
(5x + 3y + 2z)2
Solution :
(5x + 3y + 2z)2 is in the form of (a + b + c)2
Comparing (a + b + c)2 and (5x + 3y + 2z)2, we get
a = 5x
b = 3y
c = 2z
Write the formula / expansion for (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
Substitute 5x for a, 3y for b and 2z for c.
(5x + 3y + 2z)2 :
= (5x)2 + (3y)2 + (2z)2 + 2(5x)(3y) + 2(3y)(2z) + 2(5x)(2z)
(5x + 3y + 2z)2 = 25x2 + 9y2 + 4z2 + 30xy + 12yz + 20xz
So, the expansion of (5x + 3y + 2z)2 is
25x2 + 9y2 + 4z2 + 30xy + 12yz + 20xz
Example 2 :
If a + b + c = 15 , ab + bc + ac = 25, then find the value of
a2 + b2 + c2
Solution :
To get the value of (a2 + b2 + c2), we can use the formula or expansion of (a + b + c)2.
Write the formula / expansion for (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ac)
Substitute 15 for (a + b + c) and 25 for (ab + bc + ac).
(15)2 = a2 + b2 + c2 + 2(25)
225 = a2 + b2 + c2 + 50
Subtract 50 from each side.
175 = a2 + b2 + c2
So, the value of a2 + b2 + c2 is 175.
a plus b minus c Whole Square Formula
To get formula / expansion for (a + b - c)2, let us consider the formula / expansion for (a + b + c)2.
The formula or expansion for (a + b + c)2 is
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac
In (a + b + c)2, if c is negative, then we have
(a + b - c)2
In the terms of the expansion for (a + b + c)2, consider the terms in which we find "c".
They are c2, bc, ca.
Even if we take negative sign for "c" in c2, the sign of c2 will be positive. Because it has even power 2.
The terms bc, ac will be negative. Because both "b" and "a" are multiplied by "c" that is negative.
Finally, we have
(a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ac
Example :
Expand :
(x + 2y - z)2
Solution :
(x + 2y - z)2 is in the form of (a + b - c)2
Comparing (a + b - c)2 and (x + 2y - z)2, we get
a = x
b = 2y
c = z
Write the formula / expansion for (a + b - c)2.
(a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ac
Substitute x for a, 2y for b and z for c.
(x + 2y - z)2 :
= x2 + (2y)2 + z2 + 2(x)(2y) - 2(2y)(z) - 2(x)(z)
(x + 2y - z)2 = x2 + 4y2 + z2 + 4xy - 4yz - 2xz
So, the expansion of (x + 2y - z)2 is
x2 + 4y2 + z2 + 4xy - 4yz - 2xz
a minus b plus c Whole Square Formula
To get formula / expansion for (a - b + c)2, let us consider the formula / expansion for (a + b + c)2.
The formula or expansion for (a + b + c)2 is
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
In (a + b + c)2, if b is negative, then we have
(a - b + c)2
In the terms of the expansion for (a + b + c)2, consider the terms in which we find "b".
They are b2, ab, bc.
Even if we take negative sign for "b" in b2, the sign of b2 will be positive. Because it has even power 2.
The terms ab, bc will be negative. Because both "a" and "c" are multiplied by "b" that is negative.
Finally, we have
(a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ac
Example :
Expand :
(3x - y + 2z)2
Solution :
(3x - y + 2z)2 is in the form of (a - b + c)2
Comparing (a + b - c)2 and (3x - y + 2z)2, we get
a = 3x
b = y
c = 2z
Write the formula / expansion for (a - b + c)2.
(a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ac
Substitute 3x for a, y for b and 2z for c.
(3x - y + 2z)2 :
= (3x)2 + y2 + (2z)2 - 2(3x)(y) - 2(y)(2z) + 2(3x)(2z)
(3x - y + 2z)2 = 9x2 + y2 + 4z2 - 6xy - 4yz + 12xz
So, the expansion of (3x - y + 2z)2 is
9x2 + y2 + 4z2 - 6xy - 4yz + 12xz
a minus b minus c Whole Square Formula
To get the formula / expansion for (a - b - c)2, let us consider the formula / expansion for (a + b + c)2.
The formula or expansion for (a + b + c)2 is
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
In (a + b + c)2, if b and c are negative, then we have
(a - b - c)2
In the terms of the expansion for (a + b + c)2, consider the terms in which we find "b" and "c".
They are b2, c2, ab, bc, ac.
Even if we take negative sign for "b" in b2 and negative sign for "c" in c2, the sign of both b2 and c2 will be positive. Because they have even power 2.
The terms "ab" and "ac" will be negative.
Because, in "ab", "a" is multiplied by "b" that is negative.
Because, in "ac", "a" is multiplied by "c" that is negative.
The term "bc" will be positive.
Because, in "bc", both "b" and "c" are negative.
That is,
negative ⋅ negative = positive
Finally, we have
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ac
Example :
Expand :
(x - 2y - 3z)2
Solution :
(x - 2y - 3z)2 is in the form of (a - b - c)2
Comparing (a - b - c)2 and (x - 2y - 3z)2, we get
a = x
b = 2y
c = 3z
Write the formula / expansion for (a - b - c)2.
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ac
Substitute x for a, 2y for b and 3z for c.
(x - 2y - 3z)2 :
= x2 + (2y)2 + (3z)2 - 2(x)(2y) + 2(2y)(3z) - 2(x)(3z)
(x - 2y - 3z)2 = x2 + 4y2 + 9z2 - 4xy + 12yz - 6xz
So, the expansion of (x - 2y - 3z)2 is
x2 + 4y2 + 9z2 - 4xy + 12yz - 6xz
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