Reduced quadratic equations

Summary

• A reduced q uadratic equation is an equation in the form

$$x^2+px+q=0$$

• where $p, q$ are constants and $x$ is a variable.
• The discriminant of a reduced quadratic equation is defined as

$$d=p^2-4q$$

• If the discriminant is
• positive : the quadratic equation has two distin ct roots (solutions)
• zero : the quadratic equation has exactly one root (called a double root)
• negative : the quadratic equation has no real roots
• The solutions for the reduced quadratic equation when the discriminant is positive:

$$\frac{-p±\sqrt{d}}{2}$$

• When the discriminant is zero, the solution becomes $-p/2$ .
• Special cases:
• $p=0$ : the equation $x^2+q=0$ has solutions $x=±\sqrt{-q}$ for $q≤0$ .
• $q=0$ : the equation $x^2+px=0⟺x\left( x+p \right)=0$ has solutions $x=0$ , $x=-p$ .