Writing a Quadratic with Given Solutions
Not every quadratic equation can be solved by factoring, but the factoring method
can be used (in reverse) to write a quadratic equation with any given solutions. For
example, if the solutions to a quadratic equation are 5 and 3, we can reverse the
steps in the factoring method as follows:
x = 5 
or 
x 
= 3 

x  5 = 0 
or 
x + 3 
= 0 


(x  5)(x + 3) 
= 0 
Zero factor property 

x^{2}  2x  15 
= 0 
Multiply the factors. 
This method will produce the equation even if the solutions are irrational or
imaginary.
Example
Writing a quadratic given the solutions
Write a quadratic equation that has each given pair of solutions.
a) 4, 6
b)
c) 3i, 3i
Solution
a) Reverse the factoring method using solutions 4 and 6:
x = 4 
or 
x 
= 6 

x  4 = 0 
or 
x + 6 
= 0 


(x  4)(x + 6) 
= 0 
Zero factor property 

x^{2} + 2x  24 
= 0 
Multiply the factors. 
b) Reverse the factoring method using solutions
and
:
x =
x +
= 0 
or or 
x x

=
= 0 


(x +
)(x
)
x^{2}  2 
= 0 = 0 
Zero factor property Multiply the factors. 
c) Reverse the factoring method using solutions 3i and 3i:
x = 3i x + 3i = 0 
or or 
x x  3i 
= 3i = 0 


(x + 3i)(x  3i) x^{2}
 9i^{2}
x^{2} + 9 
= 0 = 0
= 0 
Zero factor property Multiply the factors
Note: i^{2} = 1 
