Quadratic Relations

By: Benjamin Wong

What is a quadratic relation?

A quadratic relation is an equation that responds to the quadratic formula. It is used to calculate height of falling rockets, thrown baseballs and even arching bridges. The quadratic formula can also be called a second degree formula.

Standard Form

A Quadratic Relation always has 2 as its highest possible exponent to have in a quadratic formula. The standard form of a quadratic relation looks like y = ax^2 + bx + c where a cannot equal to 0 however b and c can be zero.
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Common Factoring

If every term of a polynomial is divisible by the same constant the constant is called a common factor.


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

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Expanding

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.

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Factoring Trimonials

To factor trimonials you have to multiply the a value and c value from the standard form and find the factor of a and c that factors to b.
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To solve for x you have to let y = 0


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

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 = 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

Quadratic Formula is the simplest way to solve for the x value in the standard form.


The quadratic formula is

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For this example we are going to be using Solve x^2 + 3x – 4 = 0
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How we discovered the Quadratic Formula

In order to find the quadratic formula we need to start with the standard form.


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"

1. Check if Factoring is available

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