The sum of the squares of two numbers is 68 and the square of their difference is 36.
Answers
let the the no. be x & y
X² + y² = 68-------( 1 )
X² - y² = 36
(X² + y²) - 2y² = 36
∴ [ adding & subtracting " y²" ]
(68) - 2y² = 36
∴ [ From----( 1 ) ]
2y² = 68 - 36
2y² = 32
Y² = 16
Y = 4 [ ( put in eq.( 1 ) ]
X² + y² = 68
X² + 16 = 68
∴ [ y = 4 , y² = 16 ]
X² = 68 - 16
X² = 52
X = 6√2 , y = 4
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Step-by-step explanation:
The sum of the squares of two numbers is 68, and the square of their difference is 36. What is the product of the two numbers?
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The product of the two numbers is 16.
[math]x²+y²=68[/math]
[math](x-y)²=36[/math]
[math]x²-2xy+y²=36[/math]
[math]x²+y²-x²+2xy-y²=68-36[/math]
[math]2xy=32[/math]
[math]\dfrac{2xy}{2}=\dfrac{32}{2}[/math]
[math]xy=16[/math]
xy=16
x-y=6
x=6+y
[math]y(y+6)=16[/math]
[math]y²+6y-16=16-16[/math]
[math]y²+6y-16=0[/math]
[math](y-2)(y+8)=0[/math]
So y can either equal 2 or -8, and x can either equal -2 or 8.
Verifying:
[math]8²+2²[/math]
=>[math]64+4=68[/math]
[math](-8)²+(-2)²[/math]
=>[math]64+4=68[/math]
[math](8)²-2(8×2)+(2)²[/math]
=>[math]64-32+4=36[/math]
[math](-8)²-2(-8×-2)+(-2)²[/math]
=>[math]64-32+4=36[/math]
[math]xy=16[/math]
[math]-8×-2=16[/math]
[math]8×2=16[/math]
Thus the product of the numbers is 16.