# Difference between squares of two numbers is 8. Twice the square of first number by square of second number is 19...

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Difference between squares of two numbers is 8. Twice the square of first number by square of second number is 19. What are the numbers?

posted Nov 6, 2018

3 & 1

x^2-y^2=8 ------> (1)
2*x^2+y^2=19 ------> (2)

(1)+(2) => 3*x^2=27 => x^2=9 => x=3, y=1

what is wrong with this answer?

As soon as you see the phrase "difference between the squares" you should know exactly what to do. Use the difference of two squares formula:

a²-b²=(a-b)(a+b)

So we know (a+b)(a-b)=8. The only ways to factorise 8 are 1*8 and 2*4. You can't have a+b=8 and a-b=1 (since that would give 2b=9 and it's clear from context that were looking for whole numbers) so it must be a+b=4 and a-b=2. Therefore a=3 and b=1.

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