Answer
Verified
479.4k+ views
Hint: Let the smaller number be $a$,then the second number will be $a+1$. Square these and add them and then equate the sum to 313. Then solve for $a$.
Complete step-by-step answer:
Let the smaller number be $a$. Now since the numbers are consecutive, the second number will be $a+1$.
We are given that the sum of their squares is equal to 313.
${{a}^{2}}+{{\left( a+1 \right)}^{2}}=313$ …(1)
We know that ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$. We will use this to simplify ${{\left( a+1 \right)}^{2}}$
${{\left( a+1 \right)}^{2}}={{a}^{2}}+1+2a$
We will substitute this in equation (1)
${{a}^{2}}+{{a}^{2}}+1+2a=313$
$2{{a}^{2}}+2a+1=313$
$2{{a}^{2}}+2a-312=0$
Divide this by 2, we will get the following:
${{a}^{2}}+a-156=0$
This is a quadratic equation in $a$.
We know that $x=\dfrac{b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$where $a{{x}^{2}}+bx+c=0$ is a quadratic equation in $x$ .
Using this, we get the following:
$a=\dfrac{-1\pm \sqrt{{{\left( -1 \right)}^{2}}-4\times 1\times \left( -156 \right)}}{2\times 1}$
$a=\dfrac{-1\pm \sqrt{1+624}}{2}$
$a=\dfrac{-1\pm \sqrt{625}}{2}$
$a=\dfrac{-1\pm 25}{2}$
$a=\dfrac{-26}{2},\dfrac{24}{2}$
$a=-13,12$
Since, in the question we need positive integers so we will select $a=12$
So $a+1=13$
Hence, the required consecutive positive integers are $12$ and $13$.
Note: You can check whether your answer is correct or not by substituting $a=12$ and $a+1=13$ in the given equation ${{a}^{2}}+{{\left( a+1 \right)}^{2}}=313$. We get the following:
${{12}^{2}}+{{13}^{2}}=144+169=313$. So our answer is correct.
Complete step-by-step answer:
Let the smaller number be $a$. Now since the numbers are consecutive, the second number will be $a+1$.
We are given that the sum of their squares is equal to 313.
${{a}^{2}}+{{\left( a+1 \right)}^{2}}=313$ …(1)
We know that ${{\left( x+y \right)}^{2}}={{x}^{2}}+{{y}^{2}}+2xy$. We will use this to simplify ${{\left( a+1 \right)}^{2}}$
${{\left( a+1 \right)}^{2}}={{a}^{2}}+1+2a$
We will substitute this in equation (1)
${{a}^{2}}+{{a}^{2}}+1+2a=313$
$2{{a}^{2}}+2a+1=313$
$2{{a}^{2}}+2a-312=0$
Divide this by 2, we will get the following:
${{a}^{2}}+a-156=0$
This is a quadratic equation in $a$.
We know that $x=\dfrac{b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$where $a{{x}^{2}}+bx+c=0$ is a quadratic equation in $x$ .
Using this, we get the following:
$a=\dfrac{-1\pm \sqrt{{{\left( -1 \right)}^{2}}-4\times 1\times \left( -156 \right)}}{2\times 1}$
$a=\dfrac{-1\pm \sqrt{1+624}}{2}$
$a=\dfrac{-1\pm \sqrt{625}}{2}$
$a=\dfrac{-1\pm 25}{2}$
$a=\dfrac{-26}{2},\dfrac{24}{2}$
$a=-13,12$
Since, in the question we need positive integers so we will select $a=12$
So $a+1=13$
Hence, the required consecutive positive integers are $12$ and $13$.
Note: You can check whether your answer is correct or not by substituting $a=12$ and $a+1=13$ in the given equation ${{a}^{2}}+{{\left( a+1 \right)}^{2}}=313$. We get the following:
${{12}^{2}}+{{13}^{2}}=144+169=313$. So our answer is correct.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE