3Roots of Quadratics:Given a quadratic equation ax2 + bx + c the roots can be found using the quadratic formula. Three types of roots can be found:2 x-intercepts1 x-intercept (double root)No x-intercept (complex roots)Complex roots always occur in pairs. Remember: complex roots occur when b2 - 4ac < 0.
4Roots of Cubic's:Cubic equations have three roots.If we do not see three x-intercepts in the graph, then there are complex roots.Complex roots must always occur in pairs.Any graph can occur in reverse if a < 0.
53 distinct real roots2 equal, 1 distinct, real rootslocal maxlocal maxlocal minlocal min3 equal real roots2 complex, 1 real rootslocal maxNo local min/maxlocal min
6Roots of Quartic's:Quartic equations have four roots.Many possible combinations of real and complex roots.Complex roots must always occur in pairs.Any graph can occur in reverse if a < 0.
74 distinct real roots3 equal, 1 distinct, real rootslocal minlocal minabsolute minabsolute min2 equal, 2 distinct real roots2 pairs of equal real rootslocal minboth minabsolute min
84 equal real roots2 distinct real, 2 complex rootslocal minabsolute minabsolute min2 complex, 2 distinct real roots2 distinct real, 2 complex rootslocal minabsolute minabsolute min
92 equal real, 2 complex roots local minabsolute minabsolute min4 complex roots2 minima
10Graphing using RootsDetermined the roots and place the real roots on the graphDetermine if the real root crosses or touches the x-axis:Any even number of equal roots touch (not cross) the x-axis.Odd numbers of equal roots will pass through the x-axis.Place the complex roots in the graph. Complex roots may cause the graph to contain local maxima and minima. We can not yet determine their exact locations.
11Example:Find the roots of the equation0 = –(x – 2)(x2 – 5x + 10) then graph this equation.1 real x-intercept: x = 2Use the quadratic formula to find other roots:
12Since a<0 then graph starts in 2nd quadrant and ends in 4th. y-intercept: y = –(0 – 2)(02 – 5(0) + 10)y = +20(0, 20)(2, 0)