Question

How do you find the two consecutive positive integers if the sum of their square is 3445 ?

Answer 1

Hint:To solve the given question, first we need to let the two consecutive numbers. Then according to the condition given in the question, we will make the equation and simplify the given equation. We will get a quadratic equation and later we will solve the equation for the value of ‘x’. And remember that they both are positive integers and they should be consecutive.

Complete step by step answer:
We have given that,given that the sum of the squares of two consecutive numbers is 3445.Let the first number be \[x\] and the other number be\[\left( x+1 \right)\].According to the question;
\[{{x}^{2}}+{{\left( x+1 \right)}^{2}}=3445\]
Using the identity of numbers i.e. \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
Simplifying the above expression using this identity, we get
\[{{x}^{2}}+{{x}^{2}}+2x+1=3445\]
Combining the like terms, we get
\[2{{x}^{2}}+2x+1=3445\]
Subtracting 1 from both the sides of the equation, we get
\[2{{x}^{2}}+2x+1-1=3445-1\]

Simplifying the above equation, we get
\[2{{x}^{2}}+2x=3444\]
Dividing both the side of the equation by 2, we get
\[{{x}^{2}}+x=1722\]
Subtracting 1722 from both the sides of the equation, we get
\[{{x}^{2}}+x-1722=0\]
Finding the values of ‘x’,
\[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}=0\]
‘a’ is the coefficient of \[{{x}^{2}}\], ‘b’ is the coefficient of ‘x’ and ‘c’ is the coefficient constant terms. Putting all the values, we get
\[\dfrac{-1\pm \sqrt{{{1}^{2}}-4\left( 1 \right)\left( -1722 \right)}}{2\left( 1 \right)}=0\]

Simplifying the above, we get
\[\dfrac{-1\pm \sqrt{1-\left( -6888 \right)}}{2}=0\]
\[\Rightarrow \dfrac{-1\pm \sqrt{1+6888}}{2}=0\]
\[\Rightarrow \dfrac{-1\pm \sqrt{6889}}{2}=0\]
As we know that \[\sqrt{6889}\]= 83,
\[\dfrac{-1\pm 83}{2}=0\]
Thus,
We get two values,
\[\dfrac{-1+83}{2}\ and\ \dfrac{-1-83}{2}\]
Solving the above values, we get
\[\therefore\dfrac{82}{2}=41\ and\ \dfrac{-84}{2}=-42\]
Since, it is given in the question that the two numbers are positive integers.

Therefore, two numbers are 41 and 42.

Note:While solving these types of questions, students need to let the numbers by considering the conditions given in the question. As it may have seen many times they let wrong numbers and made mistakes and gave the answer as undetermined. Students need to be very careful while doing the calculation part to avoid making errors.