# Quadratic Relations (Unit 4)

### Algebra

## Topics

**What is a Quadratic Equation?****General form of a Quadratic Equation****Expanding****Factoring***Completing the square**Solving*

## 1. What is a Quadratic Equation?

__Q__**as an equation of degree 2, meaning that the highest exponent of this function is 2. Also, quadratic equations always make nice curves when you graph them.**

__uadratic Equation__## 2. General form of a Quadratic Equation

*ax2*

*+bx + c,*where

*a, b,*and

*c*are constants, and "a" can never be 0. This is not the only form that you can use to represent a quadratic equation but is the principal one, the others are the factored form [ y = a (x-r) (x-s) ] and the vertex form [ y = a (x-h)

*2 + K]*

Some examples are:* *

**1) y = 3x2 + 2x + 6**

In this one **a=3**, **b=2** and **c=6**

**2) y = x2 − 9x**

This one is a little more tricky:

Where is **a**? Well **a=1**, and we don't usually write "1x2"**b = -9**

And where is **c**? Well **c=0**, so is not shown.

**3) y = 4x − 8**

This one is **not **a quadratic equation: it is missing **x2 **

( **a=0**, which means it can't be quadratic)

## 3. Expanding

Numbers and variables satisfy an important general property, called the distributive property:

*a*(*b *+ *c*) = *ab *+ *ac*

We use the distributive property to write expressions such as 3(*x* + 1) in a form that has no brackets,

3(*x *+ 1) = 3(x) + 3(1) = 3* x *+ 3

We say that we have expanded 3(* x* + 1) to get 3

*+ 3.*

*x*To expand is to get rid of the brackets.

The distributive property says that:

*a*(*b *+ *c*) = *ab** *+ *ac*

In fact, more than this is true:

*a*(*b *+ *c *+ *d*) = *ab *+ *ac *+ *ad*

and

*a*(*b *+ *c* + *d *+ *e*) = *ab *+ *ac *+ *ad *+ *ae*

and so on.

## 4. Factoring

Examples of Simple Trinomials:

- x² + 2x + 1
- x² + 4x + 6
- x² + 3x + 6

Examples of Complex Trinomials:

- 3x² + 2x + 1
- 7x² + 4x + 4
- 5x² + 6x + 9

## 5. Completing the square

## 6. Solving

1) Factor the equation:

As shown above you can factor the equation, when you get to the factored form set each set of parenthesis equal to zero and solve for x:

One simple way of solving is by using the **quadratic formula**. With just some simple math, you will have the solution quickly. To use it you must have your equation in standard form. The quadratic formula looks like this:

Finally, the third way to solve a quadratic equation is completing the square, which is easy after you learn how to complete the square as shown above. Here are the steps: