Factoring a quadratic expression and completing the square

Summary

• A quadratic expression in \(x\) is an expression in the form

\[ax^2+bx+c\]

• where \(a,b,c\) are constants , \(a≠0\) .
• If the discriminant of the quadratic equation \(ax^2+bx+c=0\) is
• positive: then

\[ax^2+bx+c=a(x-x_1)(x-x_2)\]

• where \(x_1,x_2\) are the distinct roots of the corresponding quadratic equation
• zero : then

\[ax^2+bx+c=a{\left( x-x_1 \right)}^2\]

• where \(x_1\) is the single root of the corresponding quadratic equation
• negative : then the quadratic expression cannot be factored
• Completing the square:

\[ax^2+bx+c=a{\left( x+ \frac{b}{2a} \right)}^2+c- \frac{b^2}{4a}\]