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: ax^{2} + bx + c = 0.
The variable of the first term, ax^{2}, 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 x^{4} + 3x^{2}  10 = 0.
We can write the first term with an exponent of 2: x^{4} = (x^{2})^{2}
We can write the second term with an exponent of 1: x^{2} = (x^{2})^{1}
The third term is a constant.
To make it easier to see the quadratic form, we use the substitution u = x^{2}. That is, we replace x^{2} with u.
Original equation.

x^{4} 
+ 
3x^{2} 
 
10 
= 
0 








Think of x^{4} as (x^{2})^{2}. 
(x^{2})^{2} 
+ 
3(x^{2})^{1} 
 
10 
= 
0 








Substitute u for x^{2}. 
u^{2} 
+ 
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 = x^{2} to find the values for x.
