Solving Quadratic (2nd degree) Equations

There are 3 widely used methods for solving quadratic equations.  A quadratic (or second-degree) equation is an equation in which the variable has an exponent of 2.  The standard form of a quadratic equation is .

 The three methods used to solve quadratic equations are:  1) factoring, 2) the square root property, and 3) the quadratic formula.  Quadratic equations generally have 2 solutions.

 1)  Factoring is one method used to solve second-degree and larger equations.  First the equation must be written in standard form.  This means that the polynomial must be in descending form and set equal to zero.  Next, you must factor the polynomial.  (You may want to review factoring.)  Once the polynomial is factored, set any factor which contains a variable equal to zero and solve (using isolation) for the variable.  Check your answer in the original equation.  Every quadratic equation has two solutions, although in the case of a perfect square trinomial both of the solutions are the same.

If the equation is factored and set equal to zero as in the example, , then  and .
 
Solve the equation, .  First, you must get the equation in standard form.
This means that you must add 16 to both sides so that the  –16 is removed from the left side of the equation and the equation will then be equal to zero …  … .  Now you are ready to factor, .  Set the factors equal to zero.  Since is repeated twice as a factor, there are two solutions, but they are both the same.  Thus, is the only “unique” solution to this problem.  This is a perfect square trinomial, which factored into the square of a binomial.

If the problem has a degree of three (in other words the variable in the equation is cubed) then you will find three solutions.  Example:  .
First, factor the GCF of  +2x  from each of the terms of the polynomial … .
Next factor, .  We now have three factors which contain variables, therefore,  and  and .  This example of a third degree equation has three solutions .
 

2)  The square root property involves taking the square roots of both sides of an equation.  Before taking the square root of each side, you must isolate the term that contains the squared variable.  Once this squared-variable term is fully isolated, you will take the square root of both sides and solve for the variable.  We now introduce the possibility of two roots for every square root, one positive and one negative.  Place a  sign in front of the side containing the constant before you take the square root of that side.

Example 1:

        … the squared-variable term is isolated, so we will take the square root of
                                      each side

                    … notice the use of the  sign, this will give us both a positive and a
                                               negative root
         … simplify both sides of the equation, here x is isolated so we have
                                               solved this equation 

Example 2:

                  … again the squared-variable term is isolated, so we will take the
                                             square root of each side
         … again don’t forget the  sign, now simplify the radicals

                      … this time p is not fully isolated, also notice that 4 are
                                                         rational numbers, which means …

 and

 and

Example 3:

                  … squared term is not isolated, add 1 to each side before
                                                              beginning

        … now take the square root of both sides

              … simplify radicals

           … radical containing the constant cannot be simplified, solve for the
                                                   variable

           … notice the placement of the –1 before the radical on the
                                                            right-hand side, these numbers may not be combined since
                                                           –1 is a rational number and  are irrational numbers

 

 In each of the first 3 examples involving the square root property, notice that there were no first-degree terms.  These equations although they are quadratic in nature, have the form .  To solve a quadratic equation that contains a first-degree term using the square root property would involve completing the square which is another "trick" that will be explained in another lesson.

3)   The third method for solving quadratic equations described uses the quadratic formula.
This formula is . If you notice, the right hand side has variables a, b, and c.
These variables are the coefficients of the terms of the quadratic equation.  (Remember the standard form is .)

Example 4:

                            … first, the equation must be in standard form
                                                  …move terms to one side

                                                      … identify a = 1, b = –8, c = –9

                                             … use the quadratic formula, substitute values

                           … simplify radical

                      … solve for r

      and                    … quadratic equations have 2 solutions

Example 5:

                                               … move terms to one side and set equation equal to zero

                                         … identify a = 5, b = 1, c = –1

                                 … substitute values

                           … simplify radical

                   … cannot be simplified further

      and

Remember ... quadratic equations generally have 2 solutions.
 

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 28-Jun-2011