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655-705 Level|   Algebra|      
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Bunuel
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If x + y = 30 and x^3 + y^3 = 8100, what's the value of x^2 + y^2 ? 21 Apr 2022, 01:30
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Difficulty:

65% (hard)

Question Stats:

57% (02:48) correct 43% (02:47) wrong based on 77 sessions

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If \(x + y = 30\) and \(x^3 + y^3=8100\), what is the value of \(x^2+y^2\)?

A. \(480\)

B. \((15 + \sqrt{15})^2\)

C. \(420\)

D. \(880\)

E. Information is not sufficient to answer


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paulrahul
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If x + y = 30 and x^3 + y^3 = 8100, what's the value of x^2 + y^2 ? 22 Apr 2022, 14:11
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x3+y3 = (x+y)^3 - 3xy(x+y)
=> 8100 = (30)^3 - 3xy(30)
=> 270 = 900 - 3xy
=> 90 = 300 - xy
=> xy = 210

x2+y2 = (x+y)^2 - 2xy = 30*30 - 2*210 = 900 - 420 = 480
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If x + y = 30 and x^3 + y^3 = 8100, what's the value of x^2 + y^2 ? 31 May 2022, 11:27
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Please could someone post a full explanation, I only got as far as noting that x^2 + y^2 + 2xy = 900

But I could not work out what xy was in order to subtract 2xy from 900 to get x^2 + y^2
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If x + y = 30 and x^3 + y^3 = 8100, what's the value of x^2 + y^2 ? 01 Jun 2022, 07:20
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PVOG

I hope this is helpful. Tricky little problem that requires you to play around with the templates.

Formula 1: Sum of Cubes

(x)^3 + (y)^3 = (x + y) * (x^2 - xy + y^2)

Substitute known values ---

8100 = (30) * (x^2 - xy + y^2)

270 = x^2 - xy + y^2

(x)^2 + (y)^2 = 270 + xy (equation 1)


Formula 2: Square of a Sum

(x + y)^2 = (x)^2 + (y)^2 + 2xy

Substitute the value of (x + y) = 30
and
Substitute (equation 1) for the value of (x)^2 + (y)^2 in order to solve for (xy)

becomes -----

(30)^2 = (270 + xy) + 2xy

900 = 270 + xy + 2xy

630 = 3xy

210 = xy

xy = 210 (equation 2)


Combine (equation 1) with (equation 2) to get the value of (x)^2 + (y)^2

270 + xy = (x)^2 + (y)^2

xy = 210

270 + 210 = 480

Answer *A*
480
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Bunuel
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