SOLVING EQUATIONS BY FACTORING
(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

A cubic equation is an equation which can be written in the form ax3 + bx2 + cx + d = 0. Cubic and higher degree equations can be quite hard to factor, but textbooks and websites like ours can be kind and gentle by presenting us with special equations that are relatively easy to factor, usually using factoring by grouping. Here is an example:

 Original Equation: x3 + x2 - 9x = 9 Re-written equation: x3 + x2 - 9x - 9 = 0 Group the left-side terms: (x3 + x2) - (9x + 9) = 0 Factor the left side groups: x2(x + 1) - 9(x + 1) = 0 Factor out the common factor: (x2 - 9)(x + 1) = 0 Factor the left factor again:(x + 3)(x - 3))(x + 1) = 0 Solutions to equation: x = -3 or x = +3 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 :

 4x2 = 28x
x =
or x =

[8] Enter the roots (solutions) of the equation below :

 4x3 = 28x2
x =
or x =

[9] (edited by Ted Debiak) Enter the roots (solutions) of the equation below :

 x3 - 3x2 - 4x + 12 = 0
x =
or x =
or x =

[10] Enter the roots (solutions) of the equation below :

 2x3 + 10x2 - 2x - 10 = 0
x =
or x =
or x =

[11] Enter the roots (solutions) of the equation below :

 4x3 - 12x2 - x + 3 = 0
x =
or x =
or x =