Where's the most important substitution?

Algebra Level 3

$\large { (7+4\sqrt { 3 } ) }^{ { x }^{ 2 }-5x+5 }+{ (7-4\sqrt { 3 } ) }^{ { x }^{ 2 }-5x+5 }=14$

Suppose a 4-digit number $\overline{ABCD}$ with $A<B<C<D$ and all its 4 digits satisfy the equation above.

Determine the value of $\overline{ABCD}$ .

1 solution

Brian Charlesworth
Jun 4, 2015

Note first that $\dfrac{1}{7 - 4\sqrt{3}} = \dfrac{1}{7 - 4\sqrt{3}} \times \dfrac{7 + 4\sqrt{3}}{7 + 4\sqrt{3}} = 7 + 4\sqrt{3}.$

Then $A + \dfrac{1}{A} = 14$ for both $A = 7 + 4\sqrt{3}$ and $A = 7 - 4\sqrt{3},$ and so either

$x^{2} - 5x + 5 = 1 \Longrightarrow x^{2} - 5x + 4 = 0 \Longrightarrow x = 1,4,$ or
$x^{2} - 5x + 5 = -1 \Longrightarrow x^{2} - 5x + 6 = 0 \Longrightarrow x = 2,3.$

Thus $A = 1, B = 2, C = 3, D = 4,$ and so $\overline{ABCD} = \boxed{1234}.$

Dammit! Thought I was trying to find a 4 digit value for x. Not 4 separate solutions :(. Great solution though.

Isaac Buckley - 6 years ago

Great first observation sir :-)

Harshit Singhania - 6 years ago