(b) If $(x+y)=5$ and $x y=3$, then find the values of $x^2+y^2$ and $(x+y)^2$.
Answer (1)
Calculate ( x + y ) 2 using the given value: ( x + y ) 2 = 5 2 = 25 .
Expand ( x + y ) 2 and rearrange to find x 2 + y 2 = ( x + y ) 2 − 2 x y .
Substitute the given values into the equation: x 2 + y 2 = 25 − 2 ( 3 ) = 19 .
State the final answers: x 2 + y 2 = 19 and ( x + y ) 2 = 25 , so the values are 19 and 25 .
Explanation
Understanding the Problem We are given that ( x + y ) = 5 and x y = 3 . We need to find the values of x 2 + y 2 and ( x + y ) 2 .
Calculating ( x + y ) 2 First, let's find the value of ( x + y ) 2 . Since we know that ( x + y ) = 5 , we can simply square 5 to get ( x + y ) 2 = 5 2 = 25 .
Finding x 2 + y 2 Now, let's find the value of x 2 + y 2 . We know that ( x + y ) 2 = x 2 + 2 x y + y 2 . We can rearrange this equation to solve for x 2 + y 2 :
x 2 + y 2 = ( x + y ) 2 − 2 x y
Substituting the values We know that ( x + y ) = 5 and x y = 3 , so we can substitute these values into the equation: x 2 + y 2 = ( 5 ) 2 − 2 ( 3 ) = 25 − 6 = 19
Final Answer Therefore, x 2 + y 2 = 19 and ( x + y ) 2 = 25 .
Examples
Understanding algebraic identities like ( x + y ) 2 = x 2 + 2 x y + y 2 is useful in many real-world scenarios. For example, if you are designing a square garden with side length ( x + y ) , you might need to calculate the area of the garden, which is ( x + y ) 2 . If you later decide to divide the garden into smaller sections, you can use the identity to understand how the areas of the smaller sections relate to the total area. This type of problem also appears in physics, when calculating kinetic energy or potential energy.
