Question

The difference between squares of two numbers is 120. The square
number is twice the greater number. Find the numbers.
using only one variable
​

Answer 1

let the no. be x and2x

2x×2x-x•x=120

4x•x-x•x=120

3x•x=120

x•x=120÷3

X=√40

Answer 2

Answer:

Step-by-step explanation:

Given :

The difference of two square numbers is 120, The square is twice the greater number,.

Solution :

Let the two square numbers be x²(the smaller number) and (x² + 120) [the greater number]

Then,

$(x^2 + 120) - x^2 = 120$ (given)

⇒ $x^2 = 2 \sqrt{x^2 + 120}$ (given)

Then,

⇒ $x^4 = 4(x^2 + 120) = 4x^2 + 480$

⇒ $x^4 = 4x^2 + 480$

⇒ $x^4 - 4x^2 - 480 = 0$

⇒ $x^4 + 20x^2 - 24x^2 - 480 = 0$

⇒ $x^2(x^2  + 20) - 24(x^2 + 20) =  0$

⇒ $(x^2 + 20)(x^2 - 24)= 0$

It is possible only when one or both factors equals zero,.

⇒ x² + 20 = 0 (or) x² - 24 = 0

⇒ x² = -20 (or) x² = 24

⇒ x² = 24 (x² ≠ -20 as, square of real numbers can't be negative)

⇒ $x^2 = \sqrt{24} = 2\sqrt{6}$

The greater number ⇒ $\sqrt{x^2 + 120}=\sqrt{(\sqrt{24})^2 + 120}=\sqrt{24 + 120}= \sqrt{144} = 12$

∴ The numbers are 2√6 and 12