Solving Nonlinear Equations by Substitution
Example
Solve for w: (w^{2}  5w)^{2}  5(w^{2}  5w) = 14
Solution Step 1 Write the equation in quadratic form.
Subtract 14 from both sides.
Step 2 Use an appropriate â€œuâ€ substitution.
Substitute u for w^{2}  5w.
Step 3 Solve the resulting equation.

(w^{2}  5w)^{2}  5(w^{2}
 5w)
(w^{2}  5w)^{2}  5(w^{2}  5w)^{1} 
14
u^{2}  5u  14

= 14
= 0
= 0

Factor the new equation.
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  7)(u + 2)
u  7 = 0 or u + 2
u = 7 or u
w^{2}  5w = 7 or w^{2}  5w 
= 0
= 0
= 2
= 2 
Write each equation in standard form. Then, use
the quadratic formula to solve each equation.
The equation (w^{2}  5w)^{2}  5(w^{2}  5w) = 14 written in standard form is
(w^{2}  5w)^{2}  5(w^{2}  5w)  14 = 0. The graph of the corresponding
function, f(w) = (w^{2}  5w)^{2}  5(w^{2}  5w)  14 is shown.
The graph crosses the waxis at four locations:
