Solve Quadratic Equations Graphically

This is a tutorial on how to solve quadratic equations graphically. The quadratic equations explored are of the type

a x 2 + b x + c = 0

An applet is used to graph y = a x 2 + b x + c and change coefficients a, b and c.

Review
The solutions to the above quadratic equation are given by the quadratic formula

x1 = [ -b + sqrt(D) ] / (2a)
and
x2 = [ -b - sqrt(D) ] / (2a)

where D = b2 - 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.

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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 = b2 - 4ac
    D = b2 - 4ac = 32 - 4(2)(-5) = 49

  • Discriminant D is positive, the equation has two real solutions given by.
    x1 = [-3 + sqrt(49)] / (2*2) = 1

    x2 = [-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 = b2 - 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 = b2 - 4ac
    D = b2 - 4ac = 42 - 4(-1)(-5) = -4

  • Discriminant D is positive, the equation has two imaginary conjugate solutions given by.
    x1 = [-4 + sqrt(-4)] / (2*(-1)) = 2 - i

    x2 = [-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.

Online Step by Step Calculus Calculators and SolversNew ! Factor Quadratic Expressions - Step by Step CalculatorNew ! Step by Step Calculator to Find Domain of a Function New !
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Updated: 2 April 2013

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