Getting Started The objective is to be able to solve any quadratic equation by using the quadratic formula. Quadratic Equation - An equation in x that can be written in the standard form ax 2 + bx + c = 0, a  0. Discriminant - The expression b2 b2 - 4ac, for a quadratic equation ax 2 + bx + c = 0.

How Many Solutions Will There Be? To determine how many solutions the quadratic equation will have, determine the value of the discriminant.  If b2 b2 - 4ac is positive, then the equation has two solutions.  If b2 b2 - 4ac is zero, then the equation has one solution.  If b2 b2 - 4ac is negative, then the equation has no real solution.

How Many Solutions Are There? How many solutions do each of the following equations have? 25  169  )2)(2(  acb Since b 2 - 4ac is positive, there will be two solutions to the equation. 420  16  bac  ()()() Since b 2 –4ac is negative, there will be no real solution to the equation.

Use the quadratic formula to solve the following equation. a x bbac   x   x   68 2 x   ()()()() () x       x  x   x32 x   68 2     x  x  x  32

Determine how many solutions each equation has by using the discriminant. Use the quadratic formula to solve each equation.

16  0  bac  ()() 1644  () Since the value of the discriminant is zero, there is only one solution to this equation.

416  12  bac  ()()()  444  () Since the value of the discriminant is negative, there is no real solution to this equation.

 2   bac      16  7  123  () Because the value of the discriminant is positive, this equation will have two solutions.

            bbac a ()()()() () 111  y  5 or

     4  10  4 42   4 4   a    bbac ()() ()  or