The difference of squares of two numbers is 400. The square of the smaller number is 9 times the larger number. Find the two numbers. ​

Answers (1)

We are given that the difference of squares of  two numbers is 400. The square of the smaller number  is 9 times the larger number.

Let the larger number be   and smaller number be .

First condition states that the difference of squares of  two numbers is 400, that means;
                           x^{2} -y^{2} =400       

                   x^{2} -y^{2} -400=0              ----------- [Equation 1]

Second condition states that the square of the smaller number  is 9 times the larger number, that means;
                               y^{2} =9x

Putting this in equation 1, we get;

                   x^{2} -y^{2} -400=0            

                   x^{2} -9x-400=0 

Now, we will use middle term splitting method;

                x^{2} -25x+16x-400=0               

             x(x -25)+16(x-25)=0     

                            (x-25) (x+16)=0     

 .

Hence,   x= 25 . Since, x = -16 is not possible to take so the larger number .

Now, putting value of x in equation 1 we get;

            y^{2} = 25^{2} -400     = 625 - 400 = 225

                     y=\sqrt{225} =15

Therefore, the smaller number is y = 15.

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