3 Roots 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.
4 Roots 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.
5 3 distinct real roots2 equal, 1 distinct, real rootslocal maxlocal maxlocal minlocal min3 equal real roots2 complex, 1 real rootslocal maxNo local min/maxlocal min
6 Roots 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.
7 4 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
8 4 equal real roots2 distinct real, 2 complex rootslocal minabsolute minabsolute min2 complex, 2 distinct real roots2 distinct real, 2 complex rootslocal minabsolute minabsolute min
9 2 equal real, 2 complex roots local minabsolute minabsolute min4 complex roots2 minima
10 Graphing 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.
11 Example: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:
12 Since 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)