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
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)
solving
solving x-intercepts/roots
word problem example
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.