Question

If $ a + b = 9 $ , $ ab = 4 $ , find the value of $ {a^2} + {b^2} $

Answer 1

Hint: Let a and b are the two different variables and their sum of the two variables occurs nine and while the product of the two variables occurs four.
Square numbers are the numbers that will be multiplied to themselves with the power two; like $ {2^2} = 4 $ .
By the use of the square number’s definition and from the given two variables values while addition and multiplication we are able to find the requirement of the answer as follows.
 Formula used: $ {(a + b)^2} = {a^2} + 2ab + {b^2} $ general formula for two-variable square.

Complete step by step answer:
Since we know the formula for the two different variables $ {(a + b)^2} = {a^2} + 2ab + {b^2} $ , while using the formula $ a + b = 9 $ and $ ab = 4 $ , we will solve the given question as to the value of $ {a^2} + {b^2} $ .
We need the formula contains $ 2ab $ thus converting the given value $ ab = 4 $ into two multiplies of a and b.
Thus, we get $ 2ab = 2(4) $ (substituting the values of the product of a and b).
Further solving we get $ 2ab = 2(4) \Rightarrow 8 $ (two times the four or four times the two will occur only eight as a resultant). Hence, we know the value of $ 2ab $ .
Now on the left-hand side of the equation, there is a plus b the whole square, thus we need to convert them $ a + b = 9 $ into the whole square.
Thus, we get $ {(a + b)^2} = {9^2} $ (since squaring the value nine into two terms yields eighty-one).
Solving the above equation, we get $ {(a + b)^2} = 81 $ .
Hence substitute every know values in the general formula we get $ {(a + b)^2} = {a^2} + 2ab + {b^2} \Rightarrow 81 = {a^2} + 4 + {b^2} $ , since four can be replaced into the left side equating
Therefore, we get $ \Rightarrow {a^2} + {b^2} = 77 $ .
Hence, we get the value of $ {a^2} + {b^2} $ is seventy-seven.

Note: The general equation can be also written as in the form of $ {(a + b)^2} = (a + b)(a + b) $
While multiplying the equation into $ {(a + b)^2} = a.a + ab + ba + b.b $
Thus, this equation can be generalized into $ {(a + b)^2} = {a^2} + 2ab + {b^2} $
Hence, we must understand that this is the expression of the square values but an unknown derivative formula.