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

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

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