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Constructing a p us sized tops Equation

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If two quantities multiply to get zero, does a = 0 or b = 0? or both?

What about the following equality

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Solving for x, we get two different values : x = 2 and x = -3

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Using the FOIL method of multiplying two binomials, we obtain the function

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Relating Linear and p us sized tops

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For example:

When these expressions are expanded, they take on the General Form of quadratics:

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Forms of p us sized tops Functions

General Form

The general form for a quadratic function is given as:

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Perhaps the most important aspect of having the general form with the letter coefficients in front of the variables is finding the roots, or x intercepts, of the equation by using the tapco ar-15 handguard removal tool.

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Vertex Form

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The point (x,y) of the vertex is given by (h,k) in the equation

Where h and k can be written in terms of the coefficients a,b, and c of the General Form

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We plug the general form coefficients into our equations for h and k

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Factored Form

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Graph the quadratic function in factored form

Finding the Roots of p us sized tops

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Graphing

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The p us sized tops Formula

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The Fundamental Theorem of p us sized tops Polynomials

Suppose m and n are numbers such that

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We can multiply both sides by n and put all the terms on one side to get

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*If m and n are solutions to x2 + bx + c = 0

then

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In other words, given the general form of a quadratic equation

if a = 1,

-b = sum of the roots of the quadratic polynomial

c = product of the roots of the quadratic polynomial

which is the same as (keeping in mind that a = 1)

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For example, given the polynomial below

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For example, given the quadratic equation below

when asked to find the roots using factoring, the result would be

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Therefore, the given quadratic equation can be thought of as

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Completing the Square

Earlier we talked about square quadratic polynomials of the forms (x + α)(x + α) and how these expand to become of the general form:

The roots of the quadratic polynomial above are - and - from

which means that

The above is easy to solve, by taking the square root of both sides of the equation such that

which means that

but since a quadratic polynomial must have 2 roots, we say that

which is better written as

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is now of the same form as

and can easily be factored as

Thus we end up with

The third step is to find the square root of both sides of the equation:

to end up with

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and

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p us sized tops and Geometry

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