The Fundamentals of Quadratics

A Brief Guide to Quadratics

Solving Quadratics - Factoring

Simple Trinomials

A simple trinomial is essentially one that has 1 as the coefficient of the first term. In other words the a term in the skeleton quadratic equation (ax² + bx + c) is equal to 1. The way to solve this type of trinomial is by using the rule of product and sum.

Sum and Product Rule : find two values that when added are equal to the second term in the equation (b) and have the product equal to the third term (c).

*Also remember to take into account if the value is positive or negative*

Ex. 1 -
x ² + 5x + 6 → P: 6 S: 5


At this point we find that our values are positive 2 and 3


= (x + 2) (x + 3) → This is the final factored form for a simple trinomial

-----

Complex Trinomials

A complex trinomial is one that's first term has a coefficient (ax ² + bx + c) other than 1. This form of trinomial requires the processes of decomposition and factoring by grouping, as well as the steps used in factoring simple trinomials.


FACTORING COMPLEX TRINOMIALS

Ex. #2
6x ² - 5x - 4 → P: -24 S: -5

At this point we use decomposition, meaning we replacing our second term with our values that have a sum of -5 and product of -24


2x ² - 8x + 3x -4


Now proceed to factor by grouping so you get the following equation


2x (x - 4) + 1 (x - 4)


Finally simplify to final factored form


= (x - 4)(2x + 1)

Completing the Square

From standard form (ax ² + bx + c), completing the square can also be an option for solving the quadratic. Completing the square is often used to figure out a vertex of a parabola or for solving minimizing and optimizing questions. Once completing the square, the form you are left with will be vertex form.

Ex. # 1
(x ² + 8x) - 3 → 1) Block out the first 2 terms
2) Factor out the "a" value ( a = 1 in this case)
(x ² + 8x + 16 - 16) - 3 → 3) Divide the middle term by 2 and square it ( 8/2 = 4, 4² =16)
(x² + 8x + 16) -16 - 3 → 4) Take the negative number out (dont forget to multiply by the first term before the bracket)
(x + 4) ² -19 → 5) Middle value is divided by 2 and now the entire bracket is squared. Now left with vertex form and square is completed.
VERTEX FORM
(a - h)² + k