```Question 2592
Quadratic equations are equations where there is one unknown (x) and is in the second degree. It could also be present in the first degree. There can also be a constant (the variable is present in the 0th degree ). And as in all other "equations" the expression described above is equated to 0.
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The general form of a qudratic equation - based on the definition above is
{{{a*x^2 + b*x + c = 0}}} - where a b and c are consttants not depent on x explicitly. In some nice puzzles and problems they may be dependent on some other variable such as time etc. BUT as far this equation is conscerned they may be treated as constants (not dependent on x).

Some examples :
{{{ 2*x^2 + 3*x + 1 = 0}}}
{{{ x^2 + x + 1 = 0}}}
{{{ 3*t*x^2 + 4*t^2*x + 4/3 = 0}}}

The last example above is a quadratic in both x and t. Depending on what you want to treat as the "variable" (what you are solving for, what you wnat to obtain as a function of the otehr variable) the other can be treated as a constant.

Now, in many situations the problem may not be stated in the above form. Then the equation has to be reduced to the above form by manipulating the equation untill the right hand side becomes 0.

Take the problem of the area of the boder that was aksed on this forum some time back. It went something like "A picture has a border of constant thickness. The picture is 25 cms wide and 30 cms high. The area of the border is 174 square cms. What is the thickness of the border ? "

Draw a rectangle depicting the picture. Draw 4 lines around the rectangle each line a distance of 't' away from each face.

Then "area of the border" = "the four small squares t cms by t cmsat the corner"  + "2 rectangles t cms by 25 cm"  + "2 reactangles t cms by 30 cms".

This is expressed mathematically as
{{{ 4*t^2 + 110*t  = 174}}}.

And after converting in to into a form with 0 on the right hand side:

{{{4*t^2 +  110 * t - 174 = 0}}}.

Here t is the variable.

Now there is a standard formula applicable to all such situations which defines the "roots" of the quadratic equation.

"Root" os an equation is the value of 't' when teh left hand side also becomes zero - that is the equation is really an equation. At other values of 't' the equation will not be "balanced" - not an equation any longer. here is an example:

consider
{{{ 4*x = 20}}}.
Only when {{{ x = 5}}} the above 'equation' is 'satisfied'. At all other values of x (try for x = 4), the equation is not balanced the left hand side will not be 20. Thus the equation is said to have a 'root' at 'x=5'.

In terms of a "graph" the roots are the values of 'x' where the graph crosses the x-axis.

Now getting abck to the standard formula - I wont repeat it here - its available all over the palce. (Ok Ok I wil repeat it {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}).

Only you might be confused by the '+-' term between {{{b^2}}} and {{{sqrt(b^2 - 4*a*c) . This means that qaudratic equations (normally ) have two roots.

Let me illustrate.
Consider:
{{{ x^2 = 4}}}. It obvious that 'x=2' is a root. But why dont you consider 'x=-2'. Hmmm, thats root two. Therefore the above qduaratic has two roots. Sometimes there may be just one root. In that case the roots are said to have coincided into one value. For example consider
{{{ (x-2)^2 = 0}}} That is {{{ x^2 - 4*x + 4 = 0}}}. This has only one root - 'x = 2'.This happens because the bit under the square root sign becomes '0' for this equation. (Note that thhis is a property solely dependent on the constants of the equation.)

That means that the graph of this equation (and all such equations) just touches the x-axis and does not really cross it. For, if it crossed, it would have to cross again and tehre would be two roots.

This gives rise to the other situation where the graph may not touch teh x axis at all - ok. No roots. There can be equations which can be 'satisfied' by no value of x whatsoever.

Normally this means that the equation has no 'real' roots. No root that is a pure number. Just to address this situaitons things like imaginalry numbers have been defined. (Do a google search for this term). This happends when the term under the square root is negative. Consider
{{{x^2 + 4 = 0}}}. Now it is obvious that {{{x^2}}} is positive. Tehre fore the sum is always positive. Therefore it can never be zero. tehrefore this graph does not cross/touch x axis.

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