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Algebra AMC 10 Math Olympiad USA Math Olympiad

Problem on Equation | AMC-10A, 2007 | Problem 20

Try this beautiful problem from algebra, based on equation from AMC-10A, 2007. Problem-20,You may use sequential hints to solve the problem

Try this beautiful problem from Algebra based on quadratic equation

Problem on Equation – AMC-10A, 2007- Problem 20


Suppose that the number \(a\) satisfies the equation \(4 = a + a^{ – 1}\). What is the value of \(a^{4} + a^{ – 4}\)?

  • \(174\)
  • \(194\)
  • \(156\)

Key Concepts


Algebra

Linear equation

multiplication

Check the Answer


Answer: \(194\)

AMC-10A (2007) Problem 20

Pre College Mathematics

Try with Hints


Given that \(4 = a + a^{ – 1}\). we have to find out the value \(a^{4} + a^{ – 4}\)

Squarring both sides of \(a^{4} + a^{ – 4}\) …then opbtain…

can you finish the problem……..

\((a + a^{ – 1})^2=4^2\) \(\Rightarrow (a^2 + a^{-2} +2)=16\) \(\Rightarrow a^2 + a^{-2}=14\) and now squarring both side again………….

can you finish the problem……..

Squarring both sides of \(a^2 + a^{-2}=14\) \(\Rightarrow (a^2 + a^{-2})^2=(14)^2\) \(\Rightarrow a^4 + a^{-4} +2=196\) \(\Rightarrow a^4 + a^{-4}=194\)

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