SOLUTION: Please help me solve this equation for {{{x}}} : {{{ b^2*x^2+2*a*x = x^2 + a^2 }}} with "b doesn't equal 1". I'm really not sure where to start. Thanks so much!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Please help me solve this equation for {{{x}}} : {{{ b^2*x^2+2*a*x = x^2 + a^2 }}} with "b doesn't equal 1". I'm really not sure where to start. Thanks so much!      Log On


   

Question 662291: Please help me solve this equation for x : +b%5E2%2Ax%5E2%2B2%2Aa%2Ax+=+x%5E2+%2B+a%5E2+ with "b doesn't equal 1". I'm really not sure where to start. Thanks so much!
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
          b²x² + 2ax = x² + a²

b²x² - x² + 2ax - a² = 0

 (b²-1)x² + 2ax - a² = 0

We have to use the quadratic formula, but since the
quadratic formula contains the letters a and b, we
change the letters in the formula to capital letters:

x = %28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29+

Then A = b²-1, B = 2a, C = -a²

x = 

[Now we can see why they said "b does not equal 1" because the
denominator 2(b²-1) would be 0 if bcould equal 1, and we cannot
divide by 0].

x = %28-2a+%2B-+sqrt%28+4a%5E2-4%28-a%5E2b%5E2%2Ba%5E2%29+%29%29%2F%282%28b%5E2-1%29%29+

x = %28-2a+%2B-+sqrt%28+4a%5E2%2B4a%5E2b%5E2-4a%5E2+%29%29%2F%282%28b%5E2-1%29%29+

x = 

x = %28-2a+%2B-+sqrt%284a%5E2b%5E2+%29%29%2F%282%28b%5E2-1%29%29+

x = %28-2a+%2B-+2ab%29%2F%282%28b%5E2-1%29%29+
 
x = %282a%28-1+%2B-+b%29%29%2F%282%28b%5E2-1%29%29+

x = %28cross%282%29a%28-1+%2B-+b%29%29%2F%28cross%282%29%28b%5E2-1%29%29+

x = %28a%28-1+%2B-+b%29%29%2F%28b%5E2-1%29+

There are two solutions, one using the + and one using the -

Solution 1:


x = %28a%28-1+%2B+b%29%29%2F%28b%5E2-1%29+

Reverse the terms in the parentheses in the numerator:

x = %28a%28b+-+1%29%29%2F%28b%5E2-1%29+

Factor the denominator:

x = %28a%28b+-+1%29%29%2F%28%28b-1%29%28b%2B1%29%29+

x = %28a%28cross%28b+-+1%29%29%29%2F%28%28cross%28b-1%29%29%28b%2B1%29%29+

x = a%2F%28b%2B1%29+

Solution 2:


x = %28a%28-1+-+b%29%29%2F%28b%5E2-1%29+

Reverse the terms in the parentheses in the numerator:

x = %28a%28-b+-+1%29%29%2F%28b%5E2-1%29+

Factor -1 out of the parentheses in the numerator:

x = %28a%28-1%29%28b+%2B+1%29%29%2F%28b%5E2-1%29+

x = %28-a%28b+%2B+1%29%29%2F%28b%5E2-1%29+

Factor the denominator:

x = %28-a%28b+%2B+1%29%29%2F%28%28b-1%29%28b%2B1%29%29+

x = %28-a%28cross%28b+%2B+1%29%29%29%2F%28%28b-1%29%28cross%28b%2B1%29%29%29+

x = -a%2F%28b-1%29+

So the solutions are a%2F%28b%2B1%29+ and -a%2F%28b-1%29+

Edwin