banerjr1 wrote:
I'm getting confused on a particular step, when \(b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c\) is being simplified to \(b= \frac{c*(a+b)^2}{(a+b)^2} - c\).
When solving it I'm getting \(b= \frac{c*(a^2+b^2)}{(a+b)^2} - c\). And I'm not sure how \((a^2+b^2)\) is being written as \((a+b)^2\), as \((a+b)^2 = {a^2+2ab+b^2}\)
Let's do this with some intermediate steps so the math is more clear:
\(b= \frac{(a*\sqrt{c}+b*\sqrt{c})^2}{a^2+2ab+b^2} - c\)
Factor out \(\sqrt{c}\) from \(a*\sqrt{c}+b*\sqrt{c}\)
\(b= \frac{(\sqrt{c}*(a+b))^2}{a^2+2ab+b^2} - c\)
Distribute the square to \(\sqrt{c}\) and \(a+b\) (This is where you made your mistake — we can distribute a square to terms that are being multiplied together, but not terms that are being added together. This is why we get \((a+b)^2\) and not \(a^2 + b^2\).)
\(b= \frac{\sqrt{c}^2*(a+b)^2}{a^2+2ab+b^2} - c\)
\(b= \frac{c*(a+b)^2}{a^2+2ab+b^2} - c\)
Either expand \((a+b)^2\) in numerator to \(a^2+2ab+b^2\) OR factor \(a^2+2ab+b^2\) in denominator to \((a+b)^2\)
\(b= \frac{c*(a^2+2ab+b^2)}{a^2+2ab+b^2} - c\) OR \(b= \frac{c*(a+b)^2}{(a+b)^2} - c\)
Cancel like terms from numerator and denominator (either \(a^2+2ab+b^2\) or \((a+b)^2\))
\(b= c - c\)
\(b= 0\)
So you're totally correct that \(a^2 + b^2\) doesn't equal \((a+b)^2\), but we should never get \(a^2 + b^2\) based on exponent rules for distributing exponents.