Solving Quadratic Equations with the Quadratic Formula Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

The solutions of some quadratic equations, (), are not rational, and cannot be obtained by factoring.  For such equations, the most common method of solution is the quadratic formula.

Note:  The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored.  Be sure you know this very useful formula!!!

Examples:

1.

 By factoring (this equation is factorable): By Quadratic Formula:  a = 1,  b = 2,  c = -8 Hints: Be careful with the signs of the values a, b and c.  Don't drop the sign when substituting into the formula. Also remember your rules for multiplying and adding signed numbers as you solve the formula.

2.

 This equation cannot be solved by factoring. By Quadratic Formula:  a = 3,  b = -10,  c = 5 Hints:  Notice how the value for b was substituted into the formula using parentheses (-10).   This helps you to remember to deal with the negative value of b. Also, notice how the (-10)2 is actually a positive value.  When you square a value, the answer is always positive. If needed, these answers can be estimated as decimal values, such as (rounded to 3 decimal places): x = 2.721;        x = 0.613 The radical answers are the "exact" answers. The decimal answers are "approximate" answers.

3.

 This equation cannot be solved by factoring. By Quadratic Formula:  a = 1,  b = 4,  c = 5 Hints: Remember that a negative value under the radical is creating an imaginary number (a number with an i).

4.

 By factoring (this equation is factorable): By Quadratic Formula:  a = 1,  b = -4,  c = 4 Hints: When the value under the radical turns to zero, there will appear to be only one answer to the problem (since the plus/minus option is gone).  This really means that the one root is repeating itself, as seen in the factoring solution.

5.

 Whoa!!  Stop the presses!!! This problem cannot be solved using the Quadratic Formula until it is set equal to zero. By Quadratic Formula:  a = 2,  b = 1,  c = -1/2 Hints: