Algebra

Quadratic Expansion

 

Quadratic Expansion

 

\((a + b)^2 = a^2 + 2ab + b^2\)

\((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\)


Example 01

Expand \((2x + 5)^2\)

\begin{equation*} \begin{split} & (2x)^2 + 2 \:.\: 2x \:.\: 5 + 5^2\\\\ &  4x^2 + 20x + 25 \end{split} \end{equation*}

Example 02

Expand \((3x - 4y)^2\)

\begin{equation*} \begin{split} & (3x)^2 - 2 \:.\: 3x \:.\: 4y + (4y)^2\\\\ & 9x^2 -24xy + 16y^2 \end{split} \end{equation*}


 

Example 03

Expand \((2a + 3b + c)^2\)

\begin{equation*} \begin{split} &(2a)^2 + (3b)^2 + c^2 + 2 \:.\: 2a \:.\: 3b + 2 \:.\: 2a \:.\: c + 2 \:.\: 3b \:.\: c \\\\ &4a^2 + 9b^2 + c^2 + 12ab + 4ac + 6bc \end{split} \end{equation*}


Example 04

Expand \((m - 2n + 5r)^2\)

\begin{equation*} \begin{split} &m^2 + (-2n)^2 + (5r)^2 + 2 \:.\: m \:.\: -2n + 2 \:.\: m \:.\: 5r + 2 \:.\: -2n \:.\: 5r \\\\ & m^2 + 4n^2 + 25r^2 - 4mn + 10mr - 20nr \end{split} \end{equation*}

 

Exercise

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