and
x_{2} = [ b  sqrt(D) ] / (2a)
where D = b^{2}  4ac is called the discriminant and gives information about the number and nature of the solutions to quadratic equations. Three possibilities:
 If D > 0, the quadratic equation has 2 real solutions.
 If D = 0, the quadratic equation has 1 real solution.
 If D < 0, the equation has 2 conjugate imaginary solutions.
Interactive Tutorials
click on the button above "click here to start" to start the applet and MAXIMIZE the window obtained.
Example 1 : Solve graphically and analytically the equation
2 x ^{2} + 3x  5 = 0
Solution to Example 1:
Graphical solution:
Use the applet to set coefficients a = 2, b = 3 and c = 5 and graph the equation y = 2 x ^{2} + 3x  5. The solutions to equation 2 x ^{2} + 3x  5 = 0 correspond to points on the graph for which y = 0 which are the x intercepts: points of intersection of the graph with the x axis. These are approximately x_{ 1} = 1 and x_{ 2} = 2.5.
Analytical solution:

Given
2 x ^{2} + 3x  5 = 0

The discriminant D = b^{2}  4ac
D = b^{2}  4ac = 3^{2}  4(2)(5) = 49

Discriminant D is positive, the equation has two real solutions given by.
x_{1} = [3 + sqrt(49)] / (2*2) = 1
x_{2} = [3  sqrt(49)] / (2*2) = 2.5
The graphical and analytical solutions are equal. However in general graphical solutions are only approximate.
Example 2 : Solve graphically and analytically the equation
x ^{2} + 4x + 4 = 0
Solution to Example 2:
Graphical solution:
Use the applet to set coefficients a = 1, b = 4 and c = 4 and graph the equation y = x ^{2} + 4x + 4. There is one x intercept and the graph touches the x axis but does not cut it. These are called double or repeated soultions. x = 2
Analytical solution:

Given
x ^{2} + 4x + 4 = 0

The discriminant D = b^{2}  4ac
D = 16  4 * 4 = 0

Discriminant D is equal to zero, the equation has one double solution given by.
x = b / 2 a = 4 / 2(1) = 2
The graphical and analytical solutions are equal.
Example 3 : Solve graphically and analytically the equation
 x ^{2} + 4 x  5 = 0
Solution to Example 3:
Graphical solution:
Use the applet to set coefficients a = 1, b = 4 and c = 5 and graph the equation y =  x ^{2} + 4 x  5. There are no x intercepts and therefore the above equation has no real solutions.
Analytical solution:

Given
 x ^{2} + 4 x  5 = 0

The discriminant D = b^{2}  4ac
D = b^{2}  4ac = 4^{2}  4(1)(5) = 4

Discriminant D is positive, the equation has two imaginary conjugate solutions given by.
x_{1} = [4 + sqrt(4)] / (2*(1)) = 2  i
x_{2} = [4  sqrt(4)] / (2*(1)) = 2 + i
We cannot use the graphical method to find imaginary solutions to an equation.
Exercises: Solve graphically (using the applet) and analytically the following quadratic equations.
1: x^{ 2}  2 x = 1
2: x^{ 2} + 2 x + 10 = 0
3: x^{ 2} + 2 x = 0
Solutions to Above Exercises
1: graphical: one double solution 1 , analytical: one double solution 1
2: graphical: none , analytical: two imaginary conjugate solutions: 1  3i and 1 + 3i
3: graphical: 0 and 2 , analytical: 0 and 2
More references and links to quadratic equations.