Quadratic Equations with Imaginary Solutions
Recall that a quadratic equation is an equation that can be written in the
form ax^{2} + bx + c = 0, where a ≠ 0.
Any quadratic equation can be solved using the quadratic formula:
You probably know that if the discriminant, b^{2}  4ac, is negative then
the equation has no real number solutions.
Now we know that the square root of a negative number is an imaginary
number. In particular, if the discriminant b^{2}  4ac < 0 then the quadratic
equation has two imaginary solutions. In fact, the solutions are complex
conjugates.
Example 1
Solve using the quadratic formula: x^{2} + 2x = 5
Solution
Step 1 
Write the quadratic equation in standard form.
Add 5 to both sides of the equation. 
x^{2} + 2x = 5 
Step 2 
Identify the values of a, b, and c.
a = 1, b = 2, c = 5 

Step 3 
Substitute the values of a, b, and c
into the quadratic formula. 


Substitute 1 for a, 2 for b, and 5 for c. 

Step 4 
Simplify.
Simplify the radicand and the
denominator. 


Use
to simplify the
square root.



Cancel the common factor, 2, in the
numerator and denominator. 
x = 1 Â± 2i 
Step 5 
Check each solution.
We leave the check to you. 

So, the solutions of x^{2} + 2x = 5 are 1 + 2i and 1  2i.
Notice that 1 +2i and 1  2i are complex conjugates.
Note:
 x^{2} + 2x + 5 = 0 The discriminant of the equation is
b^{2}  4ac = 2^{2}  4(1)(5) = 16. Since the discriminant is negative, we
know that the two solutions will be
complex conjugates.

 Be careful when you cancel. Be sure to
divide each term of the numerator by 2.
