TWO METHODS OF SOLVING QUADRATIC EQUATIONS:
FACTORING
COMPLETING THE SQUARE
(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, we factor the left side into factors of the form (Dx - E). We may then conclude that x = E / D is a solution to the original equation. Here is an example:

Original Equation: 2x2 - 5x = 3
Re-written equation: 2x2 - 5x - 3 = 0
Equation with factored left side: (2x + 1)(x - 3) = 0
Solutions to equation: x = -½ or x = 3

"Completing the Square" is another method of solving ax2 + bx + c = 0.
The idea is this: re-write the equation in the form x2 + bx = c;
then add (¼)b2 to both sides, noticing that x2 + bx + (¼)b2 = (b + ½)2, a perfect square.

Original Equation: 2x2 - 8x = 10
Re-written equation: x2 - 4x = 5
Complete the left side square: x2 - 4x + 4 = 5 + 4
Equation with factored left side: (x - 2)2 = 32
Take square root of both sides: x - 2 = ± 3
Solutions to equation: x = 5 or x = -1

[1] Solve by factoring (see example above):
  x2 + 6x + 9 = 0     x =
[2] Solve by factoring (see example above):
  x2 - x - 20 = 0     x =
or x =
Review Factoring ?
[3] Solve by factoring (see example above):
  2x2 - 22x + 60 = 0     x =
or x =
Review Factoring ?
[4] Solve by factoring (see example above). Express fractions as decimals:
  2x2 + 9x = 5     x =
or x =
Review Factoring ?
[5] Solve by factoring (see example above). Express fractions as decimals:
  5x2 + 18x - 8 = 0     x =
or x =
Review Factoring ?
[6] Solve by factoring (see example above). Express fractions as decimals:
  4x2 + 6 = 11x     x =
or x =
Review Factoring ?
[7] Enter the roots (solutions) of the equation below (Methods of section 5.2 may help here):
 
If   x+2
x+3		  -  2
x		  =  5
2		   Then x =
    or x =
 
Review Factoring ?

[8] Enter the roots (solutions) of the equation below (Methods of section 5.2 may help here):
 
If   x+3
x-6		  -   x 
x-3		  =  3
4		   Then x = or x =
 
Review Factoring ?

[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 +
Solve these equations using
THE QUADRATIC FORMULA ?
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