Question

The sum of two numbers is \[25\] ,the sum of their squares is \[313\] .What are the numbers?

Answer 1

Hint: In order to solve this question, assume the two numbers as \[x\] and \[y\] .Then add the numbers and equate the sum to \[25\] and mark it as equation \[\left( i \right)\] .After that square the numbers and add then and then equate the sum to \[313\] and mark it as equation \[\left( {ii} \right)\] .And finally solve for \[x\] and \[y\].

Complete step by step answer:
Let us assume the two numbers are \[x\] and \[y\]
Now we are given that
The sum of two numbers is \[25\]
which means \[x + y = 25{\text{ }} - - - \left( i \right)\]
and the sum of their squares is \[313\]
which means \[{x^2} + {y^2} = 313{\text{ }} - - - \left( {ii} \right)\]
Now, from the equation \[\left( i \right)\] we have
\[y = 25 - x\]
Now, substitute the value of \[y\] in equation \[\left( {ii} \right)\]
Therefore, we get
\[{x^2} + {\left( {25 - x} \right)^2} = 313\]
We know that
\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
Thus, we get
\[{x^2} + \left( {{{25}^2} + {x^2} - 50x} \right) = 313\]
\[ \Rightarrow 2{x^2} - 50x + 625 = 313\]
\[ \Rightarrow 2{x^2} - 50x + 312 = 0\]
On dividing by \[2\] we get
\[ \Rightarrow {x^2} - 25x + 156 = 0\]
Now using middle term split method, we can write the above equation as
\[ \Rightarrow {x^2} - 12x - 13x + 156 = 0\]
\[ \Rightarrow x\left( {x - 12} \right) - 13\left( {x - 12} \right) = 0\]
\[ \Rightarrow \left( {x - 12} \right)\left( {x - 13} \right) = 0\]
\[ \Rightarrow \left( {x - 12} \right) = 0\] and \[\left( {x - 13} \right) = 0\]
\[ \Rightarrow x = 12\] and \[x = 13\]
Now using equation \[\left( {ii} \right)\] we find the value of \[y\] corresponding to each value of \[x\]
Therefore,
when \[x = 12\] we get
\[y = 25 - 12 = 13\]
and similarly, when \[x = 13\] we get
\[y = 25 - 13 = 12\]
So, there is only one solution which is that the two numbers are \[12\] and \[13\]
Hence, the numbers are \[12\] and \[13\] whose sum is \[25\] and the sum of squares is \[313\]

Note:
There is an alternative way to solve this question. i.e.,
Let the two numbers are \[x\] and \[y\]
Now we are given that
\[x + y = 25{\text{ }} - - - \left( i \right)\]
\[{x^2} + {y^2} = 313{\text{ }} - - - \left( {ii} \right)\]
Squaring equation \[\left( i \right)\] we get
\[{x^2} + {y^2} + 2xy = 625\]
On substituting the value from equation \[\left( {ii} \right)\] in the above equation, we get
\[313 + 2xy = 625\]
\[ \Rightarrow 2xy = 312\]
On dividing by \[2\] we get
\[xy = 156\]
Now we know that
\[{\left( {x + y} \right)^2} - {\left( {x - y} \right)^2} = 4xy\]
So, on substituting the values, we get
\[625 - {\left( {x - y} \right)^2} = 4 \times 156\]
\[ \Rightarrow {\left( {x - y} \right)^2} = 1\]
\[ \Rightarrow \left( {x - y} \right) = 1{\text{ }} - - - \left( {iii} \right)\]
Now solving equation \[\left( i \right)\] and \[\left( {iii} \right)\] using elimination method we get
\[2x = 26\]
\[ \Rightarrow x = 13\]
On substituting the value of \[x\] we get
\[y = 12\]
Hence, we get the required answer.
Also note that if you have any doubt whether the solution is correct or not, you can check it by simply substituting the values in the given conditions.