# Quadratic Relations Review

### Everything you need to know to pass the unit.

## What is a Quadratic Relation and how does one use it?

## The Three Different Forms

**Expanding**

Expanding is the opposite of Factoring. Expanding is used to rewrite polynomials with brackets and whole number powers multiplied out while all the like terms are all added together.

Ex. (*x* + *a*)(*x* + *b*) = *x*^2 + *bx* + *ax* + *ab*

= *x*^2 + *ax* + *bx* + *ab*

= *x*^2 + (*a* + *b*)*x* + *ab*

As you can see after expanding out the factored form you end up having the expanded form.

**Common Factoring**

Now Factoring is the opposite of expanding,

every term of a polynomial can be divisible by the same constant, which is called a common factor.

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

4x+20 = 4(x+5) *a* is the common factoring this equation

A polynomial is not considered factored until the Greatest Common Factor (GCF) has been factored out.

Ex.

4*x*+20 = 2(2*x*+10) NOT COMPLETELY FACTORED

4*x*+20 = 4(*x*+5) COMPLETELY FACTORED

**Standard**

*ax^*2 + *bx* + *c* is the standard form for Quadratic Relations, most people would find the X using standard form. The form is easy to use convert to expanding and factored form.

## Finding the X

**Intercepts in Vertex Form**

We are going to use y=3(x+1)^2-108

We are going to let *y* = 0

y=3(x+1)^2-108

108=3(x+1)^2

36=(x+1)^2

now the equation can go either to ways

6=(x+1)=5

or

-6=(x+1)=-7

**Quadratic Formula**

To find X, we need to do a process of isolating the X

*ax*^2 + *bx *+ *c *= 0

Now we need to complete the square.

(ax^2 + bx) + c = 0

a(x^2 + bx/a) + c = 0

a(x^2 + bx/a + b^2/4a^2 - b^2/4a^2) + c = 0

a(x^2 + bx/a + b^2/4a^2) - b^2/4a + c = 0

a(x^2 + bx/a + b^2/4a^2) - b^2/4a + 4ac/4a = 0

a(x + b/2a)^2 - b^2 + 4ac/4a = 0

After completing the square isolate x

a(x + b/2a)^2 /a = b^2 - 4ac/4a /a

√(x + b/2a)^2 = √b^2 - 4ac/4a^2

x + b/2a = +/- √b^2 - 4ac/2a

X = -b +/- √b^2 -4ac/2a- This is what you would use to find X the fastest way, and can be pretty useful when you get the hang of it. Plug in the numbers from the standard form and do the math.

## Completing the Square

Another method of solving quadratic equations is by completing the square.

Completing the square is converting the standard form of a quadratic (ax^2 + bx + c) to the vertex form *y* = *a*(*x* - *h*)^2 + *k*.

We are going to use the equation *y* = *x*2 + 2*x* + 8

1. Block off your first 2 terms: *y* = (*x*2 + 2*x* ) + 8

3. Divide the middle term by 2 then square it: (2/2)^2=1

4. Add 0: *y* = (*x*2 + 2*x* +1-1) + 8

5. Take the negative number: *y* = (*x*2 + 2*x* + 1) -1 + 8

6. Collect like terms and finish it up by factoring the polynomials: *y* = (*x* + 1)^2+ 7