Question

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. ​

Answer 1

The larger number is 25 and smaller number is 15.

Step-by-step explanation:

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 $x$  and smaller number be $y$.

• 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;

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

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

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

So, either $x-25=0$  or  $x+16=0$.

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

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.

Answer 2

Answer:

larger numbers is 25 and smaller numbers is 15