Solving Nonlinear Equations by Substitution
Example
Solve for x:
Solution 

= 18 
Step 1 Write the equation in quadratic form. 


Subtract 18 from both sides. 

= 0 
Step 2 Use an appropriate â€œuâ€ substitution. Substitute u for

u^{2} + 3u  18 
= 0 
Step 3 Solve the resulting equation.
Factor the new equation.
Use the Zero Product Property.
Solve each equation for u. 
(u + 6)(u  3)
u + 6 = 0 or u  3
u = 6 or u

= 0
= 0
= 3 
Step 4 Substitute the original
expression for u.


= 3 
Step 5 Solve for the original variable.
Multiply both sides of each
equation by x.
Write each equation in standard form. 
x^{2} + 4 = 6x or x^{2} + 4
x^{2} + 6x + 4 = 0 or x^{2}  3x + 4 
= 3x
= 0 
Neither equation factors. So, we will use the quadratic
formula to solve each:
So, there are four solutions:
The equation
written
in standard form is
The graph of the corresponding function,
is shown.
The graph crosses the xaxis at the two locations corresponding to the two
real solutions of the equation:
Note that x is in a denominator and so it cannot equal 0. Therefore, the line
x = 0 is a vertical asymptote.
