SQUARE OF A TRINOMIAL
In this section, you will learn the formula or expansion for the square of a trinomial (x + y + z). That is,
That is,
(x + y + z)2 = (x + y + z)(x + y + z)
(x + y + z)2 = x2 + xy + xz + xy + y2 + yz + xz + yz + z2
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
Example 1 :
Expand :
(5a + 3b + 2c)2
Solution :
(5a + 3b + 2c)2 is in the form of (x + y + z)2
Comparing (x + y + z)2 and (5a + 3b + 2c)2, we get
x = 5a
y = 3b
z = 2c
Write the formula / expansion for (x + y + z)2.
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
Substitute 5a for x, 3b for y and 2c for z.
(5a + 3b + 2c)2 :
= (5a)2 + (3b)2 + (2c)2 + 2(5a)(3b) + 2(3b)(2c) + 2(5a)(2c)
(5a + 3b + 2c)2 = 25a2 + 9b2 + 4c2 + 30ab + 12bc + 20ac
So, the expansion of (5a + 3b + 2c)2 is
25a2 + 9b2 + 4c2 + 30ab + 12bc + 20ac
Example 2 :
If x + y + z = 15 , xy + yz + xz = 25, then find the value of
x2 + y2 + z2
Solution :
To get the value of (x2 + y2 + z2), we can use the formula or expansion of (x + y + z)2.
Write the formula / expansion for (x + y + z)2.
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
(x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + xz)
Substitute 15 for (x + y + z) and 25 for (xy + yz + xz).
(15)2 = x2 + y2 + z2 + 2(25)
225 = x2 + y2 + z2 + 50
Subtract 50 from each side.
175 = x2 + y2 + z2
So, the value of a2 + b2 + c2 is 175.
Example 3 :
If a + b + c = 36 and a2 + b2 + c2 = 676, then find the value of (ab + bc + ca).
Solution :
To get the value of (ab + bc + ac), 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 36 for (a + b + c) and 676 for (a2 + b2 + c2).
362 = 676 + 2(ab + bc + ac)
1296 = 676 + 2(ab + bc + ac)
Subtract 676 from each side.
620 = 2(ab + bc + ac)
Divide each side by 2.
310 = ab + bc + ac
So, the value of (ab + bc + ac) is 310.
x plus y minus z Whole Square Formula
To get formula / expansion for (x + y - z)2, let us consider the formula / expansion for (x + y + z)2.
The formula or expansion for (x + y + z)2 is
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
In (x + y + z)2, if z is negative, then we have
(x + y - z)2
In the terms of the expansion for (x + y + z)2, consider the terms in which we find 'z'.
They are z2, yz, xz.
Even if we take negative sign for 'z' in z2, the sign of z2 will be positive. Because it has even power 2.
The terms yz, xz will be negative. Because both 'y' and 'x' are multiplied by 'z' that is negative.
Finally, we have
(x + y - z)2 = x2 + y2 + z2 + 2xy - 2yz - 2xz
x minus y plus z Whole Square Formula
To get formula / expansion for (x - y + z)2, let us consider the formula / expansion for (x + y + z)2.
The formula or expansion for (x + y + z)2 is
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
In (x + y + z)2, if y is negative, then we have
(x - y + z)2
In the terms of the expansion for (x + y + z)2, consider the terms in which we find 'y'.
They are y2, xy, yz.
Even if we take negative sign for 'y' in y2, the sign of y2 will be positive. Because it has even power 2.
The terms xy, yz will be negative. Because both 'x' and 'z' are multiplied by 'y' that is negative.
Finally, we have
(x - y + z)2 = x2 + y2 + z2 - 2xy - 2yz + 2xz
x minus y minus z Whole Square Formula
To get formula / expansion for (x - y - z)2, let us consider the formula / expansion for (x + y + z)2.
The formula or expansion for (x + y + z)2 is
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
In (x + y + z)2, if y and z are negative, then we have
(x - y - z)2
In the terms of the expansion for (x + y + z)2, consider the terms in which we find 'y' and 'z'.
They are y2, z2, xy, yz, xz.
Even if we take negative sign for 'y' in y2 and negative sign for 'z' in z2, the sign of both y2 and z2 will be positive. Because they have even power 2.
The terms 'xy' and 'xz' will be negative.
Because, in 'xy', 'x' is multiplied by 'y' that is negative.
Because, in 'xz', 'x' is multiplied by 'z' that is negative.
The term 'yz' will be positive.
Because, in 'yz', both 'y' and 'z' are negative.
That is,
negative ⋅ negative = positive
Finally, we have
(x - y - z)2 = x2 + y2 + z2 - 2xy + 2yz - 2xz
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