In the following window we are going to focus on the equation 3x² - 4x + 1 = 0 , which we used as an example on the previous page and gave the solutions: 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 solutions.

Numerically speaking, we noticed that the radicand of the square root of the quadratic equation formula was positive  (see the formula in Quadratic equations 1 . 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 solutions.

When you apply the quadratic equation formula you will get the "square root of 0". 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.

You can see this illustrated in the window below.

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" 

In this window we are going to focus on the equation  x² + 2x + 2 = 0

When you apply the quadratic equation formula you will get the square root of - 4 (D = - 4). "Careful" since, as you already know, the square root of a negative number cannot 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?

1.-You should notice that as the parabola does not cut the  X-axis the equation has no roots.

2.-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 does not have any roots either.

3.-Change the values of a, b and c to -2, 4 and -5 respectively and check that the parabola does not 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 Section 1.1). In this case the equation has two roots.

b) The discriminant is equal to 0 (as in Section 1. 2). In this case the equation only has one root.

c) The discriminant is a negative number (as in Section 1.3). In this case the equation does not have any roots.

a) x² - 2x 11 = 0

b) x² -1/4 = 0

c) 4x² - 4x +3 = 0

2.-Use this 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.

Biquadratic equations are those equations which relate to the fourth power and do not contain any terms of the third or first power.

Examples: x⁴ - 5x² +4 = 0 . .; . . .x⁴ - 4 = x² - 1

These equations are solved like quadratic equations to begin with. In other words, carry out the necessary operations to get rid of any denominators and bring all the terms over to the LHS and make the RHS equal to 0.

These equations can be solved graphically in the same way as the quadratic equations by drawing the graph of the LHS of the equation once the RHS is equal to 0.

Look at the graph of the first example equation x⁴ - 5x² +4 = 0 in the window below.

"Careful! From now on we are going to refer to the coefficients  x⁴, x² and the independent term as a, b and c respectively".

1.-You will notice that the graph in the window is not a parabola this time and that it cuts the X-axis at four different points!

This means, of course, that the equation has four solutions:        x = - 2 , x = - 1 , x = 1 , x = 2

2.-In your exercise book show how to solve a biquadratic equation numerically. In order to solve biquadratic equations you need to go through the following steps (as we see from the example above).

3.- Solve the following equation numerically: x⁴ - 5x² +4 = 0

a) Simplify the equation and bring all the terms over to the LHS (that's all).

b) Call x² = z  (you could use any letter) , which means that x⁴ = z²

c) Solve the equation with the new unknown factor: z² - 5z + 4 = 0  (apply the quadratic formula and you will get: z = 1 ; z = 4)

d) Use these values of z to work out the values of x: z = x² = 4 ; from which: x = the square root of 4 = ± 2

Consequently, we get the four roots which we saw earlier on the graph: x = - 2 , x = - 1 , x = 1 , x = 2

Different possible types of solutions

Bearing in mind the different types of solutions which can be obtained from a quadratic equation then we can see that biquadratic equations can have 4, 3, 2, 1, or no solutions or roots.

· Four roots when the corresponding quadratic equation has two positive roots.

· Three roots when the corresponding quadratic equation has one positive root and a 0 (the square root of 0 is 0 so therefore there is only one root).

· Two roots when the corresponding quadratic equation has a positive and a negative root (the square root of a negative number doesn't exist).

· One root when the only root given for the corresponding quadratic equation is 0 or when its roots are 0 and a negative number.

· No roots when the corresponding quadratic equation has two negative roots, just one root which is negative or no roots at all.

In this window an example is given of an equation with three roots: x⁴ - 9x² = 0

The red line represents the biquadratic graph and the blue line its corresponding quadratic graph (note that this time we have written in the complete equations in the boxes underneath).

Use the following window to solve the following biquadratic equations graphically.

Write the biquadratic equation in the box on the left and the quadratic equation which also needs solving in the other box, on the right.

a) x⁴ - 3x² + 2 = 0. 

b) x⁴ - 10x² = -9

c) x⁴ = x²

d) x⁴ - 2x² - 8 = 0