Quadratic equations

Summary

• A quadratic equation is an equation in the form

\[ax^2+bx+c=0\]

• Where \(a,b,c\) are constants, \(a≠0\) and \(x\) is a variable.
• The discriminant of a quadratic equation is defined as

\[d=b^2-4ac\]

• 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 quadratic equation when the discriminant is positive:

\[ \frac{-b±\sqrt{d}}{2a}\]

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