Quadratic relations

By.Prabhjeet

Expanding

solving expanding quadratic equations is easy if you use the distributive property to expand binomials. Collect like terms to simplify.

example 1 : ( x + 4 ) ( x – 2 )

= x2 – 2x + 4x – 8
= x2 + 2x – 8

example 2: ( x + 5 )2

= ( x + 5 ) ( x + 5 )
= x2 + 10x + 25

example 3 :( x – 3 ) ( x + 3 )

= x2 – 3x + 3x – 9
= x2 – 9
Expanding Quadratic Brackets

Factoring

Factoring is the opposite of expanding. If every term of a polynomial is divisible by the same constant, the constant is called a common factor. A polynomial is not considered to be completely factored until the greatest common factor (G.C.F. ) has been factored out.


For factoring with trinomials you just have to find two numbers whose product should give you C and added those numbers should give you B. factoring trinomial form is


Standard Form: ax^2+ bx + c


example: x^2-5X+6

x^2-3x-2x+6

x(x-3)-2(x-3

(x-3)(x-2)


example: 2y^2-2y-60

2(y^2-y-30)

2(y^2-6y+5y-30)

2[y(y-6)+5(y-6)]

2(y-6)(y+5)


Factoring when a is not 1 is simple all you have to do is

1: Multiple a by c

2: Find factors of new number

(find two numbers multiplied together to get c and added together to get b)

3: Write in standard form, using the original a and c value

4: Group and find common factors

5: Place in factor form: (x ) (x )


example:

3x^2-8x+5

3x^2-5x-3x+5

x(3x-5)-1(3x-5)

(3x-5)(x-1)

Factoring Quadratic Expressions

solving

solving for X is very easy you have to use the quadratic formula which is
Big image
Algebra Help - The Quadratic Formula - MathHelp.com

solving x-intercepts/roots

In order to solve , the first thing you will do is set both expressions that were found from factoring to make them equal to zero. Then solve for "x".

word problem example

A driver from 3m board at a swimming pool.Her height y in meters,above the water in terms of her horizontal distance x in meters, from the end of the board is given by y=-x^2+2x+3.


when does the driver hit the water?

0=x^2+2x-3

=(x-3)(x+1)


0=(x-3)=3

0=(x+1)=-1


therefore 3m

complete the square

complete the square is when we take ax^2+bx+c=0 and turn it into a(x+d)^2+e=0. Completing the square is often used to figure out a vertex of a parabola.


example: 3x^2+12x+13

3(x^2+4x)+13

3(x^2+4x+2^2-2^2)+13

3(x^2+4x+2^2)-3(4)+13

3(x+2)-12+13

3(x+2)^2+1




  • Block out the first 2 terms
  • Factor out the "a" value (which comes from standard form which we talked about earlier)
  • Take half of the x-term coefficient and square it.
  • keep the positive number and take out the negative from the bracket
  • Middle value is divided by 2 and then the entire bracket is squared.
Completing the Square - Solving Quadratic Equations