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

A Quadratic Relation is when you apply the given information into a formula in either three forms which will be talked about later. The formula is used for arcs either in charts and, structures or objects fallen in an arc. This skill is resourceful and can be used multiple ways in the real life. Not having this skill make it it harder to apply to a good job.

## 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.

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

4x+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

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.

## Factoring Trinomials

you factor trinomials by multiply the a value and c value from the standard form and find the factor of a and c that can be able to be factored into b.

<0=(x+4)(2x+3)

x+4=0

x+4-4=0-4

X+=-4

2x+3=0

2x+3-3=0-3

2x=-3

2x/2=-3/2

x = -3/2

## 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 = x2 + 2x + 8

1. Block off your first 2 terms: y = (x2 + 2x ) + 8

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

4. Add 0: y = (x2 + 2x +1-1) + 8

5. Take the negative number: y = (x2 + 2x + 1) -1 + 8

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