Solving Nonlinear Equations by Substitution
An equation that can be solved using this approach is said to be
reducible to quadratic form.
Procedure â€”
To Solve an Equation Reducible to Quadratic Form
Step 1 Write the equation in quadratic form. Step 2 Use an appropriate â€œuâ€ substitution.
Step 3 Solve the resulting equation.
Step 4 Substitute the original expression for u.
Step 5 Solve for the original variable. The letter u is traditionally used when
solving an equation by the substitution
method.
You may use a different letter if
you prefer.
Example
Solve for x: x^{4}  21x^{2}  35 = 65
Solution
Step 1 Write the equation in quadratic form.
Subtract 65 from both sides.
Write x^{4} as (x^{2})^{2}.
Step 2 Use an appropriate â€œuâ€ substitution.
Substitute u for x^{2}. 
x^{4}  21x^{2}  35
x^{4}  21x^{2}  100
(x^{2})^{2}  21(x^{2})^{1}  100
u^{2}  21u  100 
= 65
= 0
= 0
= 0 
Use the Zero Product Property.
Solve each equation for u.
Step 4 Substitute the original
expression for u.
Step 5 Solve for the original variable. 
u + 4 = 0 or u  25
u = 4 or u
x^{2} = 4 or x^{2} 
= 0 = 25 = 25 
Take the square root of both sides. 

So, there are four solutions: 2i, +2i, 5, and +5. Needless to say, these
are the same solutions that we would obtain using factoring.
Note:
The exponent of x^{4} is twice
that of x^{2}. So, we let u = x^{2}.
Then, x^{4} = (x^{2})^{2} = u^{2}.
