Factoring a reduced quadratic expression and completing the square

Summary

• A reduced q uadratic expression in \(x\) is an expression in the form

\[x^2+px+q\]

• where \(p, q\) are constants.
• If the discriminant of the reduced quadratic equation \(x^2+px+q=0\) is
• positive: then

\[x^2+px+q=(x-x_1)(x-x_2)\]

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

\[x^2+px+q={\left( x-x_1 \right)}^2\]

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

\[x^2+px+q={\left( x+ \frac{p}{2} \right)}^2+q- \frac{p^2}{4}\]