Question 248047: the difference of two positive numbers is 3 and the sum of their square is 45. Find the numbers. Found 2 solutions by stanbon, oberobic:Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website! the difference of two positive numbers is 3 and the sum of their squares is 45. Find the numbers.
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Equations:
x - y = 3
x^2 + y^2 = 45
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x = y+3
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Substitute for "x" and solve for "y":
(y+3)^2 + y^2 = 45
y^2 + 6y + 9 + y^2 = 45
2y^2 + 6y - 36 = 0
y^2 + 3y - 18 = 0
(y+6)(y-3) = 0
y = 3 or y = -6
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If y = 3, x-3 = 3, x = 6
If y = -6, x --6 = 3, x = -3
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Checking x^2 + y^2 = 45
3^2 + 6^2 = 9 + 36 = 45
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Solutions:
x=6 and y = 3
OR
x = -3 and y = -6
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Cheers,
Stan H.
You can put this solution on YOUR website! The difference of two positive numbers is 3.
x - y = 3
.
x^2 + y^2 = 45
.
We can use the first equation to solve the second.
x - y = 3
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Add y to both sides
x = y + 3
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Substitute in the second equation and solve.
x^2 + y^2 = 45
(y+3)^2 + y^2 = 45
(y^ + 6y + 9) + y^2 = 45
.
Collect terms
2y^2 + 6y + 9 = 45
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Subtract 45 from both sides.
2y^2 + 6y - 36 = 0
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Divide by 2
.
y^2 + 3y - 18 = 0
.
Can we factor 18 into two terms that are 3 apart? Yes, 3 & 6.
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(y + 6)(y - 3) = 0
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So we have two candidate answers: y = -6 and y = 3.
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The problem says the numbers are positive, so we assume y = 3 is one answer.
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Given x^2 + y^2 = 45, we know y^9 = 9.
x^2 + 9 = 45
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Subtract 9 from both sides
x^2 = 36
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Take the square root.
x = 6.
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So our second candidate answer is x = 6.
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Checking our work.
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What does x^2 + y^2 equal 45?
3^2 + 6^2 = 9 + 36 = 45
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Does x - y = 3?
6-3 = 3
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OK
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