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Home / How to Find Complex Solutions of a Quadratic Equations

How to Find Complex Solutions of a Quadratic Equations

SOLVING QUADRATIC EQUATIONS

Note:

  • A quadratic equation is a polynomial equation of degree 2.
  • The ''U'' shaped graph of a quadratic is called a parabola.
  • A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions.
  • There are several methods you can use to solve a quadratic equation:
    1. text2html_wrap_inline253 Factoring
    2. text2html_wrap_inline253 Completing the Square
    3. text2html_wrap_inline253 Quadratic Formula
    4. text2html_wrap_inline253 Graphing
  • All methods start with setting the equation equal to zero.


Solve for x in the following equation.


Example 1: text2html_wrap_inline253 tex2html_wrap_inline321

The equation is already set to zero.


Method 1: text2html_wrap_inline253 Factoring

eqnarray60

eqnarray64


Method 2: text2html_wrap_inline253 Completing the square

Divide both sides of the equation tex2html_wrap_inline323 by 2.

eqnarray80


Add tex2html_wrap_inline325 to both sides of the equation.

eqnarray97


Add tex2html_wrap_inline327 to both sides of the equation:

eqnarray121


Factor the left side and simplify the right side :

eqnarray133


Take the square root of both sides of the equation :

eqnarray141


Add tex2html_wrap_inline329 to both sides of the equation :

eqnarray150


Method 3: text2html_wrap_inline253 Quadratic Formula

The quadratic formula is tex2html_wrap_inline331

In the equation tex2html_wrap_inline333 ,a is the coefficient of the tex2html_wrap_inline335 term, b is the coefficient of the x term, and c is the constant. Substitute 2 for a, -1 for b, and -1 for c in the quadratic formula and simplify.

eqnarray189

eqnarray196


Method 4: text2html_wrap_inline253 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline341 and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline341 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

You can see from the graph that there are two x-intercepts, one at 1 and one at tex2html_wrap_inline353 .

The answers are 1 and tex2html_wrap_inline357 These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.


Check these answers in the original equation.


Check the solution x=1 by substituting 1 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  • Left Side: tex2html_wrap_inline367
  • Right Side: tex2html_wrap_inline369
Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1 for x, then x=1 is a solution.


Check the solution tex2html_wrap_inline373 by substituting tex2html_wrap_inline353 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

  • Left Side: tex2html_wrap_inline377
  • Right Side: tex2html_wrap_inline369
Since the left side of the original equation is equal to the right side of the original equation after we substitute the value tex2html_wrap_inline353 for x, then tex2html_wrap_inline373 is a solution.


The solutions to the equation tex2html_wrap_inline385 are 1 and tex2html_wrap_inline389


If you would like to work another example, click on Example.


If you would like to test yourself by working some problems similar to this example, click on Problem


If you would like to go back to the equation table of contents, click on Contents.

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Author:Nancy Marcus
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How to Find Complex Solutions of a Quadratic Equations

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