Ex 4.2 Q3 - Find two numbers whose sum is 27 and product is 182.

Ex 4.2 Q3 - Find two numbers whose sum is 27 and product is 182.

Solution:-

Let,

      First number = x

      Second number = y

Case (i): Given that, the sum of both numbers is 27,

   ∴ x + y = 27

   ∴ x = 27 - y       ------- (i)   

Case (ii): Also, the product of both numbers is 182,

   ∴ xy = 182

   ∴ (27 - y)y = 182       { from (i) }

   ∴ 27y - y² = 182

   ∴ y² - 27y + 182 = 0

   ∴ y² - 13y - 14y + 182 = 0

   ∴ y(y - 13) - 14(y - 13) = 0

   ∴ (y - 13) (y - 14) = 0

Now,

      y - 13 = 0          OR          y - 14 = 0

   ∴ y = 13                            ∴ y = 14

*Putting y = 13 in Eqⁿ(i)

   ∴ x = 27 - 13

   ∴ x = 14

*Putting y = 14 in Eqⁿ(i)

   ∴ x = 27 - 14

   ∴ x = 13

Hence,

          The required two numbers are 13 and 14.

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Try This..

✒ Two numbers differ by 5 and their product is 104. Find the numbers.

✒The sum and product of two numbers are 16 and 55 respectively. Find the numbers.

✒ The sum of two numbers is 30 and their product is 221. Find the numbers.

✒ Find two numbers whose sum is 20 and product is 96.

❌ Common Mistakes

Forgetting the signs:

▸ Always check the signs while expanding and factoring quadratic expressions.

Incorrect factor pairs:

▸ Make sure the numbers you pick for factoring actually multiply and add up to the required values.

Skipping standard form:

▸ Always write the quadratic in the ax² + bx + c = 0 format before solving.

Not verifying:

▸ Once you get the values, always plug them back to verify both the sum and product.

📝 Related Questions:-

• Ex 4.2 Q4 - Find two consecutive positive integers, sum of whose squares is 365.
• 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 Q3 Class 10

Class 10 Ex 4.2 Q3

Class 10 Chap 4 Ex 4.2 Q3

Class 10 Quadratic Equations

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