Quadratic Equations
A second degree polynomial is also called a quadratic equation or function. Examples are:
ax2+bx+c = 0 or ax2+bx+c = f(x)
A quadratic equation can have two forms. The first form is the expanded form which is the general polynomial form. The other form is factored form which is used for solving a quadratic equation. In algebra, it is important to be able to convert from one form to another.
Distribution
To go from factored form to expanded form with the polynomial, the Law of Distribution is used: a(b+c) = ab+ac. For quadratic equations, it can be more complicated because the form is usually (a+b)(c+d). Distibution in this case goes as follows:
Direct Distribution
To change the expression (a+b)(c+d) into an expanded form, the terms of one factor can be distributed to the terms of another factor:
Group Distribution
The distribution rule, a(c+d) = ac+bd, can be combined with substitution to create an intermediate step:
Foil
First multiply the first terms together then multiply the first term of the first factor with every term of the second factor. Then multiply the second term of the first factor with every term of the second factor. With quadratic equations, the distibution is called foil which means first, outside, inside and last.
Factoring
To go from expanded form to factored form, there are many methods to use. They include factoring by grouping, trial method, completing the square and the quadratic formula.
Once in factored form, the quadratic equation can be solved using the Zero Product (or Zero Factor) Theorem:
if ab = 0, then a = 0 or b = 0.
In algebra, the a and the b can represent quadratic factors. Example:
x2+4x+3 = 0 Expanded
(x+1)(x+3) = 0 Factored
When the Zero Factor Theorem is applied a = (x+1), b = (x+3) either x+1 or x+3 is equal to zero.
x+1 = 0
or
x+3 = 0
When solving for x, the result is two answers x = -1, x = -3 both of which are the solution.