Question

find two numbers whose sum is 19 if the sum of their squares is 185

Answer 1

the first number:8
the second number: 11
x+y=19
-y -y
-----------
x=19-y substitute the values in
x^2+y^2=185
(19-y)^2+y^2=185
361-38y+y^2+y^2=185
361-38y+2y^2=185
-185 -185
-----------------------------------
361-38y+2y^2-185=0 combine like terms
176-38y+2y^2=0 commutative property
2y^2-38y+176=0 divide both sides by 2
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2
y^2-19y+88=0 solve the quadratic equation
y= -(19)+square root of (-19)^2-4(1)(88)
---------------------------------------------------- simplify
2(1)
y= (19)+ square root of 361-4(88)
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2
y=19+square root of 361-352
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2
Y=19 square root of 9
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2
Y=19+3
-
------
2
y=19-3 16
---------- ---- = 8
2 2
y=19+3 22
------- ------ =11
2 2

Answer 2

The first number is 11 and the second number is 8.

Let, the first number be x and the second number be y.

The sum of two numbers whose sum is 19.

$\rm x+y=19$

The sum of their squares is 185.

$\rm x^2+y^2=185$

On solving both the equations

From equation 1

$\rm x+y=19\\\\x=19-y$

Substitute the value of x in equation 1

$\rm x^2+y^2=185\\\\(19-y)^2+y^2=185\\\\361-38y+y^2+y^2=185\\\\2y^2-38y+176=0\\\\y^2-19y+88=0\\\\y^2-11y-8y+88=0\\\\y(y-11)-8(y-11)=0\\\\(y-11)(y-8)=0\\\\y-11-0, \ y=11\\\\y-8=0, \ y=8$

Substitute y = 11 in equation 1

$\rm x+y=19\\\\x+11=19\\\\x=19-11\\\\x=8$

Substitute y = 8 in equation 1

$\rm x+y=19\\\\x+8=19\\\\x=19-8\\\\x=11$

Hence, the first number is 11 and the second number is 8.

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