Quadratic equations.
An explanation of different cases
 
Different types of solutions (an explanation of different types of roots)

1) EQUATIONS WITH TWO SOLUTIONS

Exercise 1.- The equation: 3x2 - 4x + 1 = 0 , which we used as an example in the exercises in the first part of this unit: (Quadratic equations 1 ), gave the roots: x = 1 and x = 1/3.

Remember that in this case the graph obtained from the equation (a parabola) cut the X-axis at two points. Therefore, the equation had two roots.

Look carefully at the graph of the equation in this window.

Numerically speaking, we said that the radicand of the square root of the quadratic equation formula was positive (see quadratic equation formula in "general solution" section). This value is called the "discriminant" of the equation. It is usually represented by a small triangle but we are going to call it "D" instead.
In this example the discriminant D = 16 - 12 = 4 > 0. Therefore, the equation has two roots.

2) EQUATIONS WITH ONLY ONE SOLUTION

Exercise 2.- Solve the following equation in your exercise book:

x2 - 2x +1 = 0

After applying the quadratic equation formula you will have got the "square root of 0" as an answer. Therefore, the "discriminant of the equation is 0".

What does this mean?

As the square root of 0 is 0, the "only solution" we get of the equation is x = 2/2 = 1. Therefore, in this case there is only one root of the equation.
In this window you can see how the parabola only cuts the X-axis at one point, when x = 1. Therefore:

"If a quadratic equation only has one root then the parabola cuts the X-axis at one single point, which is the vertex of the parabola" 

"If you change the value of the parameters and want to go back to the original values just click on the 'Init' button".
3) EQUATIONS WITH NO SOLUTION

Exercise 3.- Solve the following equation in your exercise book:

x2 + 2x + 2 = 0

After applying the quadratic equation formula you will have got the square root of - 4 (D = - 4). "Careful" since, as you already know, the square root of a negative number can't exist.

Therefore, we can say that in this case the equation has no solution.

However, let's see what this means from a graphical point of view:

Look carefully at this window where the values of "a", "b" and "c" are 1, 2 and 2 respectively.

What do you notice now about the parabola with respect to the X-axis?

Obviously, as the parabola does not cut the  X-axis then the equation has no roots.

Exercise 4.- Solve the equation: -2x2 +4x - 5= 0

Change the value of the parameters in the window. Be careful with the signs! You should get the square root of -24 as an answer. Therefore, this equation doesn't have any roots either.

Change the values of a, b and c to -2, 4 and -5 respectively and check that the parabola doesn't cut the X-axis in this case either.

Therefore:

"If a quadratic equation doesn't have any roots then the corresponding graph does not cut the X-axis".

TO SUM UP

We have seen that the number of roots of a quadratic equation depends on the sign of the number which we get inside the square root of the quadratic equation formula. In other words, the sign of the "discriminant" of the equation which is equal to:

D = b2 - 4ac. The following situations are all possible:

a) The discriminant is a positive number (as in Exercise 1). In this case the equation has two roots.

b) The discriminant is equal to 0 (as in Exercise 2). In this case the equation only has one root.

c) The discriminant is a negative number (as in Exercises 3 and 4). In this case the equation doesn't have any roots.

Exercises
Exercise 5 .- Use this window to solve the following equations graphically by changing the parameters a, b and c appropriately.
a) x2 - 2x 11 = 0
b) x2 -1/4 = 0
c) 4x2 - 4x +3 = 0

Now use the quadratic equation formula to solve these same equations numerically, in your exercise book, and check that the solutions are the same.

Exercise 6.- Use the window to find quadratic equations (whose coefficients are whole numbers) which are different to those you have just solved, which have two, one or no roots and make a note of what these roots are.

Practise as much as you want by changing the equations that appear in the window.
Exercise 7.- In your exercise book write down at least two equations of each type by working out the value of the "discriminant" and seeing how it corresponds to the number of roots of the equation in each case.
  Volver al índice   Atrás   adelante  
           
  Leoncio Santos Cuervo
 
© Spanish Ministry of Education and Science. Year 2001