How to Solve Quadratic Equations Having Complex Solutions?

This blog introduces an expansion of the real number system into the Complex Number System. It covers operations with complex numbers and finding complex solutions to quadratic equations.

In the series of free math tutorials on solving quadratic equations, it’s our final blog on teaching students the complex solutions. Now once again the underlying assumption is that you have read our previous blogs on solving quadratic equations on the following methods:

  1. Factor method 
  2. Completing the square
  3. Graphing

So, let’s switch gears and get started with exploring, learning and having fun with complex solutions.

 

Let’s first understand what do we mean by a complex number?

A complex number is a combination of real values and imaginary values. It is denoted by z = a + ib, where a, b are real numbers and i is an imaginary unit. i=-1and no real value satisfies the equationi2= -1, therefore, the term having i is called the imaginary part of the solution..

Problems which previously had no solution can be solved with the inclusion of the complex number system.

Eager to learn how to find complex solutions?

Let’s solve the following equations having complex number solutions and master the art of solving quadratic equations.

To solve the quadratic equations having complex solutions, we need to compare the given equation with the standard form of quadratic equation, which is ax2+bx+c=0. And, after identifying the constants, we will be using the quadratic formula to find the solution. Here is the quadratic formula:

Eager to learn how to find complex solutions


x=
-bb2-4ac2a

 

Example 1: x2=3x-5

Step 1: Rewrite the given quadratic equation in standard form, and identify the constants. 

x2=3x-5

x2-3x+5=0

After comparing with the standard form of quadratic equation ax2+bx+c=0, we get a=1, b=-3, and c=5.

Step 2: Apply the quadratic formula.

x=-bb2-4ac2a

x=-3-32-41521

x=9-202

x=-112

x=11i2

Step 3: Separate the solutions. 

x1=3+i112

x2=3-i112

 

Example 2: 2x2-6x+5=0

Step 1: Compare the given quadratic equation with the standard form of the quadratic equation and identify the constants.  

2x2-6x+5=0

After comparing with the standard form of quadratic equation ax2+bx+c=0, we get a=2, b=-6, and c=5.

Step 2: Apply the quadratic formula.

x=-bb2-4ac2a

x=-6-62-42522

x=36-404

x=-44

x=4i4=6±2i4

x=23±i4=3±i2

Step 3: Separate the solutions. 

x1=3+i2

x2=3-i2

 

Example 3: 8x2-4x+5=0

Step 1: Compare the standard form of the quadratic equation with the given equation and identify the constants.  

8x2-4x+5=0

After comparing with the standard form of quadratic equation ax2+bx+c=0, we get a=8, b=-4, and c=5.

Step 2: Apply the quadratic formula.

x=-bb2-4ac2a

x=-4-42-48528

x=16-16016

x=-14416

x=144i16=4±12i16

x=41±3i16=1±3i4

Step 3: Separate the solutions. 

x1=1+3i4

x2=1-3i4

We hope these steps were easy to follow and you understood them well. 

Here are a few practice problems for you to solve if you feel confident about the solutions to the above problems using complex numbers.

  • -7x^2 + 12x = 10 
  • 5x^2 + 8x = -4
  • 5x^2 + 8x + 5 = 0
  • -5x^2 + 12x – 8 = 0 
  • –x^2 + 4x – 5 = 0


Algebra 2 is an important part of the high school syllabus, and solving simple and complex quadratic equations is a part of the school curriculum. Understanding all four methods of solving quadratic equations not only enables you to score well in class and maintain a good GPA score but it also enables you to achieve a good score on your SAT exam, which we all know is an important part of a college application process. 

And if you are one of the students who simply loves math and Algebra 2 then, you can make use of this knowledge and pursue a math undergraduate degree and go on to even make a good career out of it.

In this blog, we talked about complex numbers and how high school students solve quadratic equations with complex solutions. 

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