Quadratic Expansion
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*}
\((a + b + c\dotso)^2 = \text{square of each elements + two times of each two elements}\)
Proof:
\begin{equation*} \begin{split} (a + b + c)^2 & = (a + b + c)(a + b + c)\\\\ (a + b + c)^2& = a(a + b + c) + b(a + b + c) + c(a + b + c)\\\\ (a + b + c)^2& = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2\\\\ (a + b + c)^2& = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \end{split} \end{equation*}
Example 1
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 2
Expand \((m - 2n + 5r)^2\)
\begin{equation*} \begin{split} & (m + (-2n) + 5r)^2\\\\ &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*}
Factorize \(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\)
\(\color{blue}1^{st}\) step: root each element which has power of two
\(a^2 \rightarrow \sqrt{a^2} = (a) \text{ or } (-a)\)
\(b^2 \rightarrow \sqrt{b^2} = (b) \text{ or } (-b)\)
\(c^2 \rightarrow \sqrt{c^2} = (c) \text{ or } (-c)\)
\(\color{blue}2^{nd}\) step: look at the two times of each two elements
\(2ab \rightarrow 2(a)(b) \text{ or } 2(-a)(-b)\)
\(2ac \rightarrow 2(a)(c) \text{ or } 2(-a)(-c)\)
\(2bc \rightarrow 2(b)(c) \text{ or } 2(-b)(-c)\)
\(\color{blue}3^{rd}\) step: There are two option for the answer
\(\color{purple} 1^{st}\text{ option}\)
\(\text{The positive element } a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = (a + b + c)^2\)
\(\color{purple} 2^{nd}\text{ option}\)
\(\text{The negatif element } a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = (-a - b - c)^2\)
Example
Factorize \(x^2 + y^2 + 16 + 2xy + 8x + 8y\)
\(\color{blue}1^{st}\) step: root each element which has power of two
\(x^2 \rightarrow \sqrt{x^2} = x\)
\(y^2 \rightarrow \sqrt{y^2} = y\)
\(16 \rightarrow \sqrt{4^2} = 4\)
\(\color{blue}2^{nd}\) step: look at the two times of each two elements
\(2xy \rightarrow 2(x)(y)\)
\(8x \rightarrow 2(x)(4)\)
\(8y \rightarrow 2(4)(y)\)
\(\color{blue}3^{rd}\) step: \(x^2 + y^2 + 16 + 2xy + 8x + 8y = (x + y + 4)^2\)
Exercise