SQUARE OF A TRINOMIAL –
Let us find the square of the trinomial x + y + z
So, (x + y + z)² = {(x + y) + z }²
= {(x + y)² + 2.(x + y).z + z²}
[Applying the formula (a+b)²= a²+2ab+b²]
= (x² + 2xy + y²) + 2xz + 2yz + z²
[Applying the formula (a+b)²= a²+2ab+b²]
= x²+ 2xy + y² + 2xz + 2yz + z²
= x²+ y²+ z² + 2(xy + yz + zx)
There are some example are given below for your better understanding –
Example.1) (2a + 4b + 6c)²
Ans.) (2a + 4b + 6c)²
= {(2a + 4b) + 6c}²
= (2a + 4b)²+ 2 X 6c X (2a + 4b) + 36c²
= 4a²+ 16b²+ 2 X 2aX 4b + 12c (2a + 4b) + 36c²
= 4a²+ 16b²+ 16ab + 24ac + 48bc + 36c²
= 4 (a²+ 4b²+ 9c²+ 4ab + 6ac + 12bc) (Ans.)
Example.2) (3x + 5y - 7z)²
Ans.) (3x + 5y - 7z)²
= {(3x + 5y) - 7z}²
= (3x + 5y)²- 2 X (3x + 5y) X 7z + 49z²
= 9x²+ 2 X 3x X 5y + 25y²- 42xz – 70yz + 49z²
= 9x²+ 25y²+ 49z²+ 30xy – 42xz – 70yz (Ans.)
Example.3) (2p – 3q + 4r)²
Ans.) (2p – 3q + 4r)²
= {(2p – 3q) + 4r}²
= {(2p – 3q)²+ 2 X (2p – 3q) X 4r + (4r)²}
= (4p²- 2 X 2p X 3q + 9q²) + 16pr – 24qr + 16r²
= 4p²- 12pq + 9q² + 16pr – 24qr + 16r²
= 4p² + 9q² + 16r² - 12pq + 16pr – 24qr (Ans.)
Example.4) (2a – 3x – 5p)²
Ans.) (2a – 3x – 5p)²
= {2a – (3x + 5p)}²
= 4a²- 2 X 2a X (3x + 5p) + (3x + 5p)²
= 4a²- 12ax – 20ap + ( 9x² + 2 X 3x X 5p + 25p²)
= 4a²- 12ax – 20ap + 9x²+ 30 xp + 25p²
= 4a²+ 9x²+ 25p²- 12ax – 20ap + 30xp (Ans.)
OR,
(2a – 3x – 5p)²
= {(2a – 3x) – 5p}²
= (2a – 3x)²- 2 X (2a – 3x) X 5p + 25p²
= 4a²- 2 X 2a X 3x + 9x²- 20ap + 30xp + 25p²
= 4a²+ 9x²+ 25p²- 12ax – 20ap + 30xp (Ans.)
Corollaries –
Two corollaries follow from this expansion
1) (a + b + c)² - 2(ab + bc + ca)
= a² + b² + c² + 2(ab + bc + ca) - 2(ab + bc + ca)
= a² + b² + c²
So, a²+ b²+ c² = (a + b + c)² - 2(ab + bc + ca)
2) (a + b + c)² - (a² + b² + c²)
= a²+ b²+ c²+ 2(ab + bc + ca) - (a² + b² + c²)
= 2(ab + bc + ca)
So, 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
There are some examples are given below for your better understanding -
Example.1) If a + b + c = 10 and ab + bc + ca = 20, find a²+ b²+ c²
Ans.) As per the given condition, a + b + c = 10 and ab + bc + ca = 20, we have to find out the value of a²+ b²+ c² = ?
As per the formula, 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
So, (a² + b² + c²) = (a + b + c)² - 2(ab + bc + ca)
= 10² - (2X20) (putting the value)
= 100 – 40 = 60
So, (a²+ b²+ c²) = 60 (Ans.)
Example.2) If, a + b + c = 15 and a²+ b²+ c² = 125, then find ab + bc + ca = ?
Ans.) As per the given condition, a + b + c = 15 and a²+ b²+ c² = 125, we have to find out the value of ab + bc + ca = ?
As per the formula, 2(ab + bc + ca) = (a + b + c)² - (a² + b² + c²)
2(ab + bc + ca) = 15² - 125 (putting the value)
2(ab + bc + ca) = 225 - 125 = 100
ab + bc + ca = 100/2 = 50 (Ans.)