Solving Nonlinear Equations by Substitution
Some nonlinear equations can be rewritten so that they can be solved using
the methods for solving quadratic equations.
Recall the general form of a quadratic equation: ax2 + bx + c = 0.
The variable of the first term, ax2, has an exponent of 2.
The variable of the second term, bx, has an exponent of 1.
The third term, c, is a constant.
If we can rewrite an equation in quadratic form then we can solve the
equation by using a method for solving a quadratic equation, such as
factoring or by using the quadratic formula.
For example, consider the equation x4 + 3x2 - 10 = 0.
We can write the first term with an exponent of 2: x4 = (x2)2
We can write the second term with an exponent of 1: x2 = (x2)1
The third term is a constant.
To make it easier to see the quadratic form, we use the substitution u = x2. That is, we replace x2 with u.
| Original equation.
|
x4 |
+ |
3x2 |
- |
10 |
= |
0 |
| |
 |
|
 |
|
 |
|
 |
| Think of x4 as (x2)2. |
(x2)2 |
+ |
3(x2)1 |
- |
10 |
= |
0 |
| |
 |
|
 |
|
 |
|
 |
| Substitute u for x2. |
u2 |
+ |
3u |
- |
10 |
= |
0 |
The last equation is in quadratic form. We can solve it by factoring or by
using the quadratic formula.
After solving for u, we can use u = x2 to find the values for x.
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