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 =

[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 (Methods of section 5.2 may help here):

 If x+2x+3 - 2x = 52 Then x =     or x =

[8] Enter the roots (solutions) of the equation below (Methods of section 5.2 may help here):

 If x+3x-6 - x x-3 = 34 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 +
 Solve these equations using THE QUADRATIC FORMULA ?