Equations in a

Question

1. Expand (a - 1/a)²

2. If a > 1 and a² + 1/a² = 706/225, find the value of

i) a - 1/a

ii) a + 1/a

iii) a

Answer 1

1.   (a-1/a)² = (a-1/a)(a-1/a) = a² -a(1/a)-a(1/a)+(1/a)(1/a) = a²-2+1/a²

 

2.   a>1,  a²+1/a²=706/225

 

Note that it will be easier to first find the value of 'a' then the subsequent values.

   

iii.    Multiply through by a²

       a²(a²+1/a²)=706a²/225

       a⁴ + 1 = 706a²/225

       a⁴ -(706/225)a²+1=0

       Let x=a² to turn this into a quadratic equation

       x² - (706/225)x +1 = 0

 

       From this point we can use the quadratic equation to solve for x

    x = [-b ±√(b²-4ac)]/2a   where a=1, b = -706/225, c=1

    x = [706/225 ±√[(-706/225)²-4(1)(1)]/2(1)

    x = 706/225(2) ±√((498436/50625)-4)/2

    x = 706/450 ±[√((498436-202500)/50625)]/2

    x = 706/450 ±[√(295936/50625)]/2

    x ≅ 1.569 ±1.209

    x ≅ 2.778  or  1.667

    Remember that x = a²

    a² ≅ 2.778  or  .36

    a ≅ √2.778  or  √.36

    a ≅ 1.667  or  .6

 

Given the condition that a>1 our answer is  a≅ 1.667

 

Check:  a²+1/a²=706/225

           1.667²+ 1/1.667² = 706/225

           3.139 = 3.138

Given the necessary rounding this answer is correct.

 

i.  a-1/a = 1.667 - 1/1.667 ≅ 1.067

 

ii.  a+1/a = 1.667 + 1/1.667 ≅ 2.267