Quadratic Relations
By: Benjamin Wong
What is a quadratic relation?
Standard Form
Common Factoring
Ex: ab+ac = a(b+c) where a is the common factor.
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
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
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
Therefore x = -4 or x = 3/2
Completing the square
We are going to use the equation y = x^2 + 8x - 3
1. Block off your first 2 terms: y = (x^2 + 8x) - 3
2. Factor out the a value (we don't have an a value but if we did it would look like this: (2x^2 + 4x) = 2(x^2 + 2x)
3. Divide the middle term by 2 then square it: (8x/2)^2 = 16
4. Add 0: y = (x^2 + 8x +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 = 4x^2 + 24x + 36
0 = (2x - 5) (2x - 7)
x = 5/2 or x = 7/2
Quadratic Formula
The quadratic formula is
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