The Square of The Sum of Two Expressions
\((a + b)^2 = a^2 + 2ab + b^2\)
The Square of The Difference of Two Expressions
\((a - b)^2 = a^2 - 2ab + b^2\)
Example 1
Expand \((2x + 5)^2\)
\begin{equation*}
\begin{split}
(2x + 5)^2 &= (2x)^2 + 2(2x)(5) + 5^2\\\\
(2x + 5)^2 & = 4x^2 + 20x + 25
\end{split}
\end{equation*}
Example 2
Expand \((3x - 4y)^2\)
\begin{equation*}
\begin{split}
(3x - 4y)^2 &= (3x)^2 - 2(3x)(4y) + (4y)^2\\\\
(3x - 4y)^2& = 9x^2 -24xy + 16y^2
\end{split}
\end{equation*}
Example 3
Factorize \(x^2 + 6x + 9\)
\begin{equation*}
\begin{split}
x^2 + 6x + 9& = x^2 + 2\cdot 3\cdot x + 3^2\\\\
x^2 + 6x + 9& = x^2 + 2\cdot x \cdot 3 + 3^2\\\\
x^2 + 6x + 9& = (x + 3)^2
\end{split}
\end{equation*}
Example 4
Factorize \(x^2 - 10xy + 25y^2\)
\begin{equation*}
\begin{split}
x^2 - 10xy + 25y^2& = x^2 - 2\cdot x\cdot 5y + 5^2y^2\\\\
x^2 - 10xy + 25y^2& = x^2 - 2\cdot x\cdot 5y + (5y)^2\\\\
x^2 - 10xy + 25y^2& = (x - 5y)^2
\end{split}
\end{equation*}