Let $a = b$.

Then,
$\begin{array} {c r c l }
\color{#D61F06} {1} & ab & = & a^2 \\
\color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\
\color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\
\color{#20A900} {4} & b & = & a+b \\
\color{#3D99F6} {5} & b & = & b+b \\
\color{#302B94} {6} & b & = & 2b \\
\color{#69047E} {7} & 1 & = & 2.
\end{array}$
Which line is the **first** that contains an error in this proof?

$\begin{array} {c r c l } \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \end{array}$

On the left-hand side of line 3, the value $b$ is "factored out" from $ab-b^2$ to get $b(a-b) .$

Is the equation below true? $ab - b^2 - a^2 = ab(1-b-a)$

$\begin{array} {c r c l } \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \end{array}$

Changing $a^2 - b^2$ to $(a+b)(a-b)$ is known as the difference of squares identity (it was first introduced in a previous course, as shown in the video below). What if we started with fourth powers instead, that is, $a^4 - b^4 ?$ Which expression is equivalent?