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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

But try the problem first…

Answer: $194$

Source

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 $a^2 + a^{-2}=14$ $\Rightarrow (a^2 + a^{-2})^2=(14)^2$ $\Rightarrow a^4 + a^{-4} +2=196$ $\Rightarrow a^4 + a^{-4}=194$