Quadratic Equation by Middle Term Breaking

Example 1

$$\textup{Solve } x^2+9x+20=0$$

Solution

\begin{align*} x^2+9x+20 & = 0\\ x^2+5x+4x+20 & = 0\\ x(x+5)+4(x+5) & = 0\\ (x+5)(x+4) & = 0\\\\ x+5=0, \,\,\,\, & x+4=0\\ x=0-5, \,\,\,\, & x=0-4\\\\ x=-5, \,\,\,\, & x=-4\\ \end{align*}

Example 2

$$\textup{Solve } x^2+12x+35=0$$

Solution

\begin{align*} x^2+12x+35 & = 0\\ x^2+5x+7x+35 & = 0\\ x(x+5)+7(x+5) & = 0\\ (x+5)(x+7) & = 0\\\\ x+5=0, \,\,\,\, & x+7=0\\ x=0-5, \,\,\,\, & x=0-7\\\\ x=-5, \,\,\,\, & x=-7\\ \end{align*}

Example 3

$$\textup{Solve } x^2+2x-15=0$$

Solution

\begin{align*} x^2+2x-15 & = 0\\ x^2+5x-3x-15 & = 0\\ x(x+5)-3(x+5) & = 0\\ (x+5)(x-3) & = 0\\\\ x+5=0, \,\,\,\, & x-3=0\\ x=0-5, \,\,\,\, & x=0+3\\\\ x=-5, \,\,\,\, & x=3\\ \end{align*}

Example 4

$$\textup{Solve } x^2+4x-32=0$$

Solution

\begin{align*} x^2+4x-32 & = 0\\ x^2+8x-4x-32 & = 0\\ x(x+8)-4(x+8) & = 0\\ (x+8)(x-4) & = 0\\\\ x+8=0, \,\,\,\, & x-4=0\\ x=0-8, \,\,\,\, & x=0+4\\\\ x=-8, \,\,\,\, & x=4\\ \end{align*}