Find two consecutive positive integers, sum of whose squares is 365 ?

byAmlesh Mahato •March 04, 2021

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Solution:

⇨ Let the two consecutive positive integers are x, x+1

A/Q

$Find two consecutive positive integers, sum of whose squares is 365.$

$\large \Rightarrow x^2+x^2+2x+1=365$

$\large \Rightarrow 2x^2+2x+1-365=0$

$\large \Rightarrow 2x^2+2x-364=0$

⇨Divided by 2 on both side

$\large \Rightarrow x^2+x-182=0$

$\large \Rightarrow x^2+(14-13)x-182=0$

$\large \Rightarrow x^2+14x-13x-182=0$

$\large \Rightarrow x(x+14)-13(x+14)=0$

$\large \Rightarrow (x-13)(x+14)=0$

$\large \Rightarrow x-13=0,x+14=0$

$\large \Rightarrow x=13,x=-14$

⇨x = -14, We only consider positive integers, so -14 is rejected.

⇨Therefore, Two consecutive positive integers are = x , x+1

⇨13 , 13+1

⇨13, 14

Exercise 4.2

1. Find the roots of the following quadratic equations by factorisation:

(i) $x^2-3x-10=0$

(ii) $Find the roots of 2x^2+x-6=0 quadratic equations by factorization ?$

(iii) $Find the roots of \sqrt{2}x^2+7x+5\sqrt{2}=0 quadratic equations by factorization ?$

(iv) $Find the roots of 2x^2-x+\frac{1}{8}=0 quadratic equations by factorization ?$

(v) $\large 100x^{2}-20x+1=0$

3. Find two numbers whose sum is 27 and product is 182.

4. Find two consecutive positive integers, sum of whose squares is 365.

5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ` 90, find the number of articles produced and the cost of each article.

Tags: Chapter-4 Exercise-4.2 Math- Class 10

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