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 $d= \frac{b^2-4ac}{a^2}$ .