Quadratic equations.
Biquadratic equations
 
Biquadratic equations

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

For example: 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.

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

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

Find the roots by dragging the red point or by changing the x value in the box in the lower part of the window.

Numerical solutions

In order to solve biquadratic equations you need to go through the following steps (as we saw in the example above).

Exercise 8.- 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 = 1 ; from which: x = the square root of 1 = ± 1 or

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 - 4x2 = 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).

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Exercise 9.- Solve the equation above x4 - 4x2 = 0 numerically and check that the solutions match those given in the graph.

Exercise 10.- Use this window to solve the following biquadratic equations:

a) x4 - 3x2 + 2 = 0
b) x4 - 10x2 = -9
c) x4 = x2
d) x4 - 2x2 - 8 = 0

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.

Solve these equations numerically and check that the solutions match those given in the graph.

"You'll have to look carefully at the points where the graphs cut the X-axis. When the values are not whole numbers click with the mouse on top of the point to see the approximate values given as its coordinates".
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  Leoncio Santos Cuervo
 
© Spanish Ministry of Education and Science. Year 2001