SOLVING QUADRATIC EQUATIONS
THE QUADRATIC FORMULA
(self test)

A quadratic equation is an equation which can be written in the form ax2 + bx + c = 0 where a, b, and c are constants (as opposed to x, which is a variable). To solve an equation such as ax2 + bx + c = 0 by factoring, or by by completing the square, check out these links.

For quadratic equations which are difficult to factor, the Quadratic Formula will always reveal solutions. Traditionally, these solutions are written:

x =
-b ±

 b2 - 4ac
2a
Here is an example:

Original Equation: 2x2 + 5x = 10
Re-written equation: 2x2 + 5x - 10 = 0
Identify values of "a", "b", and "c": a = 2, b = 5, c = -10
Substiture these values into the quadratic formula, to get:
x =
-(5) ±

 (5)2 - 4(5)(-10)
2(2)
  =   - 5 ± 15
4		   =  
-5
5
2

In other examples, b2-4ac might be negative; in this case, x may be imaginary or complex.
[1] Solve using the quadratic formula (see example above).
Express real number answers as a decimal accurate to within 0.01, such as "1.33"
Express imaginary (complex) answers in the form a + bi and a - bi, such as "1 + 3i":
  x2 + 3x - 9 = 0     x =     or     x =
[2] Solve by factoring (see example above):
  x2 - x - 20 = 0     x =
or x =
[3] Solve by factoring (see example above):
  2x2 - 22x + 60 = 0     x =
or x =
[4] Solve by factoring (see example above). Express fractions as decimals:
  2x2 + 9x = 5     x =
or x =
[5] Solve by factoring (see example above). Express fractions as decimals:
  5x2 + 18x - 8 = 0     x =
or x =
[6] Solve by factoring (see example above). Express fractions as decimals:
  4x2 + 6 = 11x     x =
or x =
[7] Enter the roots (solutions) of the equation below (Review Algebra ?):
 
If   x+2
x+3		  -  2
x		  =  5
2		   Then x =
    or x =
 

[8] Enter the roots (solutions) of the equation below (Review Algebra ?):
 
If   x+3
x-6		  -   x 
x-3		  =  3
4		   Then x = or x =
 

[9] Enter the number or decimal which completes the following square:
  x2 + 6x +
[10] Enter the number or decimal which completes the following square:
  x2 - 3x +
[11] Enter the number or decimal which completes the following square:
  x2 - (½)x +
[12] Enter the number or decimal which completes the following square:
  x2 - 5x +
[13] Enter the number or decimal which completes the following square:
  x2 - x +
[14] Enter the number or decimal which completes the following square:
  x2 + x +
   
This page last
updated 20 Aug 2000