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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
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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.
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