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

But try the problem first…

Answer: \(194\)

AMC-10A (2007) Problem 20

Pre College Mathematics

## Try with Hints

First hint

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……..

Second Hint

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

can you finish the problem……..

Final Step

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

## Other useful links

- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=afpsj0gqHfU

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