help with this inequality algebra 2 problem please help by tonight

Question

the sum of two positive numbers equals the difference of the squares of the two numbers which equals the quotient of the larger number when divided by the smaller. what is the smaller number.

the equation that i got was a+b=a^2 - b^2=a/b

Answer 1

Your equations are right. Let's use a+b=a/b. Rewrite as: ab+b^2=a, so b^2+ab=a, so a(1-b)=b^2. For a to be positive b must be between 0 and 1 or 0<b<1. a^2-b^2=(a-b)(a+b)=a+b, because the difference of the squares is equal to a+b. So dividing both sides by a+b we get a-b=1, so a=1+b. Therefore substituting for a in a(1-b)=b^2 we get 1-b^2=b^2, and 2b^2=1, from which b=sqrt(1/2) or sqrt(2)/2 and a=1+sqrt(2)/2. Therefore approximately b=0.7071 and a=1.7071. The smaller number is 0.7071.

Comments

thank you so much