Difference between revisions of "2020 AMC 10A Problems/Problem 14"
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As above, we need to calculate <math>\frac{(x^2+y^2)(x^3+y^3)}{x^2y^2}</math>. Note that <math>x,y,</math> are the roots of <math>x^2-4x-2</math> and so <math>x^3=4x^2+2x</math> and <math>y^3=4y^2+2y</math>. Thus <math>x^3+y^3=4(x^2+y^2)+2(x+y)=4(20)+2(4)=88</math> where <math>x^2+y^2=20</math> and <math>x^2y^2=4</math> as in the previous solution. Thus the answer is <math>\frac{(20)(88)}{4}=\boxed{\textbf{(D)}\ 440}</math>. | As above, we need to calculate <math>\frac{(x^2+y^2)(x^3+y^3)}{x^2y^2}</math>. Note that <math>x,y,</math> are the roots of <math>x^2-4x-2</math> and so <math>x^3=4x^2+2x</math> and <math>y^3=4y^2+2y</math>. Thus <math>x^3+y^3=4(x^2+y^2)+2(x+y)=4(20)+2(4)=88</math> where <math>x^2+y^2=20</math> and <math>x^2y^2=4</math> as in the previous solution. Thus the answer is <math>\frac{(20)(88)}{4}=\boxed{\textbf{(D)}\ 440}</math>. | ||
− | \textbf{- Emathmaster} | + | <math>\textbf{- Emathmaster}</math> |
==Video Solution== | ==Video Solution== |
Revision as of 13:49, 1 February 2020
Real numbers and satisfy and . What is the value of
Contents
Solution
Continuing to combine From the givens, it can be concluded that . Also, This means that . Substituting this information into , we have . ~PCChess
Solution 2
As above, we need to calculate . Note that are the roots of and so and . Thus where and as in the previous solution. Thus the answer is .
Video Solution
~IceMatrix
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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