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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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When these expressions are expanded, they take on the General Form of quadratics:
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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 who checks for carbon monoxide.
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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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Graph the quadratic function in factored form
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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
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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 -
which means that
The above is easy to solve, by taking the square root of both sides of the equation
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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