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 \(x=0\) , \(x=-p\) .