Introduction to sum of two squares
In algebra, we have some formulae to expand squares.
They are:
`( a + b ) ^2 = a^2 + 2ab + b^2`
`( a ** b ) ^2` = `a^2 ** 2ab + b^2`
`( a + b ) ^2 + ( a ** b ) ^2` = `2 ( a^2 + b^2 )`
`( a + (1/a) ) ^2` = `a ^2 + (1/a^2) + 2`
`( a ** 1/a ) ^2` = `a ^2 + 1/a ^2 ** 2`
`( a + (1/a) ) ^2` + `( a ** 1/a ) ^2` = `2 ( a ^2 + (1/a ^2))`
`a + b = sqrt (( a ** b ) ^2 + 4 ab)`
`a ** b ` = `sqrt (( a + b ) ^2 ** 4 ab)`
Keeping these formulae in mind, we can break up the square values to get the final answer. Now let us see few problems on sum of two squares. I like to share this Example of Obtuse Angle with you all through my article.
Example Problems on Sum of Two Squares
Ex 1: Find the value of a ^2 + b^2, If a + b = 7 and ab = 7.
Soln: By using the above formulae, `a^2 + b ^2 = (1/2) [ ( a + b ) ^2 + ( a ** b ^2]`
Therefore a – b = `sqrt (( a + b ) ^2 **4 ab)`
a – b = `sqrt (7 ^2 ** 4 ( 7 ))` = `sqrt (49 **28)` = `sqrt 21`
Therefore `a ^2 + b ^2` = `(1/2)[ ( a+ b ) ^2 + ( a ** b) ^2]`
= `(1/2) [ 7^2 + ( sqrt21)^2]` = `(1/2) [ 49 + 21 ]`
Therefore `a^2 + b^2 = 35`
Ex 2: Find the value of A^2 + b^2, if a – b = 7 and ab = 18.
Soln: By using the above formula, `a + b = sqrt (( a ** b) ^2 + 4 ab)`
= `sqrt ((7) ^2 + 4 (18)) = sqrt (49 + 72)`
= `sqrt 121` = 11
Therefore `a^2 + b^2` = `(1/2) [( a + b ) ^2 + ( a ** b ) ^2]`
= `(1/2) [ 11^2 + 7 ^2 ] = (1/2)[ 121 + 49 ]`
= `(1/2)` [ 170 ] = 85
Therefore `a^2 + b^2 = 85`
Ex 3: If `a + (1/a) = 6` , find the value of `a ^2 + (1/a^2)` .
Soln: `a ^2 + (1/a^2) = (a + (1/a) ) ^2 **2 = ( 6 ^2) ** 2 = 34`
Therefore `a^2 + (1/a^2) = 34` [By using formula in 4]
Ex 4: If `a ** (1/a) = 8` , find the value of `a^2 + (1/a ^2)`
Soln: Therefore `a^2 + 1/a^2` = `(a ** (1/a)) ^2` + 2
= `8^2 + 2`
= 64 + 2 = 66.
Therefore `a^2 + (1/a^2)` = 66. [By using formula in 5]
Ex 5: If a^2 – 5a – 1 = 0, find the value of `a^2 + (1/a^2)`
Soln: Given: a2 – 5a – 1 = 0
`rArr` a – 5 – (1/ a) = 0 [Divide throughout by]
`rArr` `a **(1/a)` = 5
Therefore `a^2 + (1/a^2)` = `(a ** (1/a) ) ^2` + 2 = `5^2` + 2 = 27. Understanding Area of Hexagon is always challenging for me but thanks to all math help websites to help me out.
Practice Problems on Sum of Two Squares
1. If a + 1/a = 2, find a^2 + 1/a^2
Ans: 2
2. If a + b = 9 and ab = -22, find the values of a ^2 + b^2.
[And: 125]
3. If a^2 – 3a + 1 = 0, find the value of a^2 + 1/a ^2.
[Ans: 7]
In algebra, we have some formulae to expand squares.
They are:
`( a + b ) ^2 = a^2 + 2ab + b^2`
`( a ** b ) ^2` = `a^2 ** 2ab + b^2`
`( a + b ) ^2 + ( a ** b ) ^2` = `2 ( a^2 + b^2 )`
`( a + (1/a) ) ^2` = `a ^2 + (1/a^2) + 2`
`( a ** 1/a ) ^2` = `a ^2 + 1/a ^2 ** 2`
`( a + (1/a) ) ^2` + `( a ** 1/a ) ^2` = `2 ( a ^2 + (1/a ^2))`
`a + b = sqrt (( a ** b ) ^2 + 4 ab)`
`a ** b ` = `sqrt (( a + b ) ^2 ** 4 ab)`
Keeping these formulae in mind, we can break up the square values to get the final answer. Now let us see few problems on sum of two squares. I like to share this Example of Obtuse Angle with you all through my article.
Example Problems on Sum of Two Squares
Ex 1: Find the value of a ^2 + b^2, If a + b = 7 and ab = 7.
Soln: By using the above formulae, `a^2 + b ^2 = (1/2) [ ( a + b ) ^2 + ( a ** b ^2]`
Therefore a – b = `sqrt (( a + b ) ^2 **4 ab)`
a – b = `sqrt (7 ^2 ** 4 ( 7 ))` = `sqrt (49 **28)` = `sqrt 21`
Therefore `a ^2 + b ^2` = `(1/2)[ ( a+ b ) ^2 + ( a ** b) ^2]`
= `(1/2) [ 7^2 + ( sqrt21)^2]` = `(1/2) [ 49 + 21 ]`
Therefore `a^2 + b^2 = 35`
Ex 2: Find the value of A^2 + b^2, if a – b = 7 and ab = 18.
Soln: By using the above formula, `a + b = sqrt (( a ** b) ^2 + 4 ab)`
= `sqrt ((7) ^2 + 4 (18)) = sqrt (49 + 72)`
= `sqrt 121` = 11
Therefore `a^2 + b^2` = `(1/2) [( a + b ) ^2 + ( a ** b ) ^2]`
= `(1/2) [ 11^2 + 7 ^2 ] = (1/2)[ 121 + 49 ]`
= `(1/2)` [ 170 ] = 85
Therefore `a^2 + b^2 = 85`
Ex 3: If `a + (1/a) = 6` , find the value of `a ^2 + (1/a^2)` .
Soln: `a ^2 + (1/a^2) = (a + (1/a) ) ^2 **2 = ( 6 ^2) ** 2 = 34`
Therefore `a^2 + (1/a^2) = 34` [By using formula in 4]
Ex 4: If `a ** (1/a) = 8` , find the value of `a^2 + (1/a ^2)`
Soln: Therefore `a^2 + 1/a^2` = `(a ** (1/a)) ^2` + 2
= `8^2 + 2`
= 64 + 2 = 66.
Therefore `a^2 + (1/a^2)` = 66. [By using formula in 5]
Ex 5: If a^2 – 5a – 1 = 0, find the value of `a^2 + (1/a^2)`
Soln: Given: a2 – 5a – 1 = 0
`rArr` a – 5 – (1/ a) = 0 [Divide throughout by]
`rArr` `a **(1/a)` = 5
Therefore `a^2 + (1/a^2)` = `(a ** (1/a) ) ^2` + 2 = `5^2` + 2 = 27. Understanding Area of Hexagon is always challenging for me but thanks to all math help websites to help me out.
Practice Problems on Sum of Two Squares
1. If a + 1/a = 2, find a^2 + 1/a^2
Ans: 2
2. If a + b = 9 and ab = -22, find the values of a ^2 + b^2.
[And: 125]
3. If a^2 – 3a + 1 = 0, find the value of a^2 + 1/a ^2.
[Ans: 7]
