Ch.1 Quadratics

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A quadratics equation takes the form

\[ax^2+bx+c=0\]

where \(a,b,c\) are constants, \(a\neq 0\)

This equation is interesting because if we were to plot the equation as a function \(y\) of \(x\), we get an parabola. The graph, as we shall see, is charateristic of the motion exhibited by an object under the influence of a gravitational force.  We shall examine the physical use of such graphs in due course, but for now, let us focus on the equation itself.

Solving Quadratic Equations

It should come as no surprise that equations have solutions, and quadratic equations are no exception. The solution of a quadratic equation is when the graph of the equation intersects the x-axis, or in other words, its function value is 0. This value is also referred to as the root, or  zero of the quadratic.

\[f(\text{solution})=0\]

We can solve quadratic equations in different ways, depending on what form of the quadratic you are dealing with. The easiest of which is:

Factorisation

The factorisation method uses the Null Factor law, where

\[\text{If }ab=0\ text{then }a=0\text{ or }b=0\]

This is really nothing but common sense. If two number are multiplied together and their product is 0, then logic dictates either one, or both of them are zero. Since we are looking for the roots of the quadratic equation, this will be quite handy. Let \(f(x)\) be

\[x^2-3x-4=0\]

We want to factorise the equation such that the equation takes the form:

\[(x-a)(x-b)=0\text{, where a, b, are roots of the quadratic}\]


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