# Quadratic Relations

### By: Benjamin Wong

## Standard Form

*ax^*2 +

*bx*+

*c*where

*a*cannot equal to 0 however

*b*and

*c*can be zero.

## Common Factoring

Ex: *ab*+*ac* = ** a**(

*b*+

*c*) where

**is the common factor.**

*a*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**

## Expanding

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.

## Factoring Trimonials

*a*value and

*c*value from the standard form and find the factor of

*a*and

*c*that factors to

*b*.

*y*= 0

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

*x* + 4 = 0

*x* + 4 - 4 = 0 - 4

*X* + = -4

2*x* + 3 = 0

2*x* + 3 - 3 = 0 - 3

2*x* = -3

2*x*/2 = -3/2

*x* = -3/2

Therefore *x* = -4 or *x* = 3/2

## Completing the square

*y*=

*a*(

*x*-

*h*)^2 +

*k*.

We are going to use the equation *y* = *x*^2 + 8*x* - 3

1. Block off your first 2 terms: *y* = (*x*^2 + 8*x*) - 3

2. Factor out the a value (we don't have an a value but if we did it would look like this: (2*x*^2 + 4*x*) = 2(*x*^2 + 2*x*)

3. Divide the middle term by 2 then square it: (8*x*/2)^2 = 16

4. Add 0: *y* = (*x*^2 + 8*x* +16 - 16) - 3

5. Take the negative number: y = (x^2 + 8x + 16) - 16 - 3

6. Collect like terms: y = (x^2 + 8x + 16) - 19

7. Finish it up by factoring the polynomials: y = (x + 4)^2 - 19

## Finding the X - Intercepts in Vertex Form

We are going to use *y* = 4(x - 3)^2 - 1

We are going to let *y* = 0

0 = 4(*x* - 3)^2 - 1

0 = 4(*x*^2 + 6x + 9) - 1

0 = 4*x*^2 + 24x + 36

0 = (2*x* - 5) (2x - 7)

*x* = 5/2 or *x* = 7/2

## Quadratic Formula

*x*value in the standard form.

The quadratic formula is

*x*^2 + 3

*x*– 4 = 0

## How we discovered the Quadratic Formula

*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**

## Which Formula to use when solving for "x"

ie. x^2 + 2x - 3

2. If factoring is not available see if the "a" value is 1 or is able to be square root. If yes try completing the square.

ie. 4x^2 - 8x - 1 = 0

3. If none of the above works, use the quadratic formula X = -b +/- √b^2 -4ac/2a