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 (≠ 0), 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 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 = 25
Re-written equation: 2x2 + 5x - 25 = 0
Identify values of "a", "b", and "c": a = 2, b = 5, c = -25
Substiture these values into the quadratic formula, to get:
x =
-(5) ±

 (5)2 - 4(2)(-25)
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 (as in [2] below).
[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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 + 3x - 9 = 0     x =     or     x =
[2] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 + 2x + 2 = 0     x =     or     x =
[3] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 + x = 3     x =     or     x =
[4] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 - 5x + 5 = 0     x =     or     x =
[5] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x(x + 6) + 13 = 0     x =     or     x =
[6] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 + 2x + = 0 Then x = or x =

[7] Solve using the quadratic formula (Methods of section 5.2 may help here).
Express real number answers as a decimal accurate to within 0.01, such as "1.33"
Express imaginary answers as a+bi and a-bi without spaces, such as "1+3i":
 
If   x+2
x+3		  -  2
x		  =  3   Then x = or x =
 

[8] Solve using the quadratic formula (Methods of section 5.2 may help here).
Express real number answers as a decimal accurate to within 0.01, such as "1.33"
Express imaginary answers as a+bi and a-bi without spaces, such as "1+3i":
 
If   x+3
x-6		  -   x 
x-3		  = 1   Then x = or x =
 

[9] 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 answers as a+bi and a-bi without spaces, such as "1+3i":
  x2 + 1 = 0   Then x = or x =

[10] 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 answers as a+bi or a-bi without spaces, such as "1+3i":
  x2 + 2ix - 5 = 0   Then x = or x =

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