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 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.
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.
4x+20 = 2(2x+10) NOT COMPLETELY FACTORED
4x+20 = 4(x+5) COMPLETELY FACTORED
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
now the equation can go either to ways
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 = 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