Equations Quadratic in Form
In a quadratic equation we have a variable and its square (x and x^{2}). An equation
that contains an expression and the square of that expression is quadratic in form
if substituting a single variable for that expression results in a quadratic equation.
Equations that are quadratic in form can be solved by using methods for quadratic
equations.
Example
An equation quadratic in form
Solve (x + 15)^{2}  3(x + 15)  18 = 0
Solution
Note that x + 15 and (x + 15)^{2} both appear in the equation. Let a
= x + 15 and
substitute a for x + 15 in the equation:
(x + 15)^{2}
 3(x + 15)  18 
= 0 

a^{2}  3a  18 
= 0 

(a  6)(a + 3) 
= 0 
Factor. 
a  6 
= 0 
or 
a + 3 
= 0 

a 
= 6 
or 
a 
= 3 

x + 15 
= 6 
or 
x + 15 
= 3 
Replace a by x + 15. 
x 
= 9 
or 
x 
= 18 

Check in the original equation. The solution set is {18, 9}.
