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 \(a^2 + a^{-2}=14\) \(\Rightarrow (a^2 + a^{-2})^2=(14)^2\) \(\Rightarrow a^4 + a^{-4} +2=196\) \(\Rightarrow a^4 + a^{-4}=194\)
Other useful links
- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=afpsj0gqHfU