Ex 4.2 Q4 - Find two consecutive positive integers, sum of whose squares is 365.

Ex 4.2 Q4 - Find two consecutive positive integers, sum of whose squares is 365.

Solution:-

Let,

      First integer = x

   ∵ Numbers are consecutive positive

       integers,

   ∴ Second integer = x + 1

➙ It is given that, the sum of the squares of first and second integer is 365,

   ∴ (First integer)² + (Second integer)² = 365

   ∴ (x)² + (x + 1)² = 365

   *Using formula, (a + b)² = a² + 2ab + b²

   ∴ x² + x² + 2x + 1 = 365

   ∴ 2x² + 2x = 365 - 1

   ∴ 2x² + 2x = 364

   *Taking 2 as common,

   ∴ x² + x = 182

   ∴ x² + x - 182 = 0

   *Splitting the middle term, we get,

   ∴ x² + 14x - 13x - 182 = 0

   ∴ x(x + 14) - 13(x + 14) = 0

   ∴ (x + 14) (x - 13) = 0

Now,

      x + 14 = 0         OR          x - 13 = 0

   ∴ x = -14                          ∴ x = 13

   *Since, we have to find consecutive positive integers,

   ∴ x = -14 is not possible

   ∴ x = 13

Therefore,

   *First integer

   = x

   = 13

   *Second integer

   = x + 1

   = 13 + 1

   = 14

Hence,

         The required consecutive positive integers are 13 and 14.

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Try This..

✒ If the square of a number is 7 more than five times the number, find the number.

✒ Find two consecutive numbers whose product is 182.

✒ The sum of the squares of two consecutive numbers is 421. Find the numbers.

✒ Find two consecutive positive odd integers whose squares add up to 394.

❌ Common Mistakes

Not simplifying:

▸ Forgetting to simplify the equation before solving (like dividing by 2).

Choosing the wrong values:

▸ After solving the quadratic equation, remember, the question asks for positive integers.

Errors in expanding (x + 1)²:

▸ Always double-check your binomial expansions!

Not verifying:

▸ Not checking your answer by substituting back into the original question.

📝 Related Questions:-

• Ex 4.2 Q5 - The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
• Ex 4.2 Q6 - 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.

Queries Solved:-

Class 10 Ex 4.2

Ex 4.2 Q4 Class 10

Class 10 Ex 4.2 Q4

Class 10 Chap 4 Ex 4.2 Q4

Class 10 Quadratic Equations

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