# 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

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

__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 +__5__x +__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

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## 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__and__**decomposition****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 -

**8**

**x + 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

**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**