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Quadratic Equations

A park is in the shape of a right triangle. The length of the hypotenuse and a side of this park are 25 m and 7 m respectively. What is the length of the third side?

Let us consider the following triangle ABC as the right triangular park whose hypotenuse AC = 25 m and side BC = 7 m.

Let AB be x.

Using Pythagoras theorem for triangle ABC,

AB2 + BC2 = AC2

x2 + (7)2 = (25)2

x2 + 49 = 625

x2 = (625 − 49)

x2 = 576 m2

To find the value of x, we require a number whose square is 576.

For this, we will find the square root of 576. Mathematically, we write it as

Here,  represents the square root. To find the square root of 576, we follow a method, which is known as prime factorization method.

Let us discuss this method and find the square root of 576 with the help of the given video.

In this way, we can find the square root of a given number by prime factorization method and solve problems related to it.

Let us discuss some more examples to understand the concept better.

Example 1:

Find the square roots of the following numbers.

(i) 324

(ii) 676

(iii) 1225

(iv) 3136

Solution:

(i) The prime factorization of 324 is

(ii) The prime factorization of 676 is

(iii) The prime factorization of 1225 is 

(iv) The prime factorization of 3136 is  

Example 2:

Is 504 a perfect square? If not, 

(i) find the smallest number multiplied to this number, so that the product would be a perfect square

(ii) find the smallest number by which 504 must be divided, so that the quotient is a perfect square

Solution:

The prime factorization of 504 is

Here, the prime factors 2 and 7 do not occur in pair. Therefore, 504 is not a perfect square.

(i) In order to obtain a perfect square, each factor of 504 must be paired. Therefore, we have to make pairs of 2 and 7. For this, 504 should be multiplied by 2 × 7 i.e., 14.

Therefore, 14 should be multiplied with 504 to make it a perfect square.

(ii) In order to obtain a perfect square, each factor of 504 must be paired. Therefore, 504 should be divided by 2 × 7 i.e., 14.

Therefore, 504 should be divided by 14 so that the quotient is a perfect s…

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