Problem: Understanding the quadratic formula

Problem

  1. Show that

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

  1. Show that ( \(d\) is the discriminant, \(d=p^2-4q\) )

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

  1. Show that \(x^2+px+q=0\) can have no solutions when \(d<0\) .
  2. Show that \(x^2+px+q=0\) will have precisely one solution \(x=-p/2\) when \(d=0\) .
  3. Show that \(x^2+px+q=0\) will have two solutions, \(x= \frac{-p±\sqrt{d}}{2}\) when \(d>0\) .
  4. Show that \(x^2+px+q={\left( x-x_1 \right)}^2\) when \(d=0\) .
  5. Show that \(x^2+px+q=\left( x-x_1 \right)\left( x-x_2 \right)\) when \(d>0\) .
  6. Show that \(ax^2+bx+c=0\) iff \(x^2+ \frac{b}{a}x+ \frac{c}{a}=0\) .
  7. Show that the discriminant of \(x^2+ \frac{b}{a}x+ \frac{c}{a}=0\) is \(d= \frac{b^2-4ac}{a^2}\) .

Solution