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Example: Solving the quadratic equation. Suppose `ax^2+bx+c=0` and `a!=0`. We first divide by `a` to get `x^2+b/ax+c/a=0`. Then we complete the square and obtain `x^2+b/ax+(b/(2a))^2-(b/(2a))^2+c/a=0`. The first three terms factor to give `(x+b/(2a))^2=(b^2)/(4a^2)-c/a`. Now we take square roots on both sides and get `x+b/(2a)=+-sqrt((b^2)/(4a^2)-c/a)`. Finally we move the `b/(2a)` to the right and simplify to get the two solutions: `x_(1,2)=(-b+-sqrt(b^2 - 4ac))/(2a)`

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Example: Solving the quadratic equation.
Suppose `ax^2+bx+c=0` and `a!=0`. We first divide by `a` to get `x^2+b/ax+c/a=0`. 

Then we complete the square and obtain `x^2+b/ax+(b/(2a))^2-(b/(2a))^2+c/a=0`. 
The first three terms factor to give `(x+b/(2a))^2=(b^2)/(4a^2)-c/a`.
Now we take square roots on both sides and get `x+b/(2a)=+-sqrt((b^2)/(4a^2)-c/a)`.

Finally we move the `b/(2a)` to the right and simplify to get 
the two solutions: `x_(1,2)=(-b+-sqrt(b^2-4ac))/(2a)`

Here are a few more examples:

Type this	See that	Comment
\`x^2+y_1+z_12^34\`	`x^2+y_1+z_12^34`	subscripts as in TeX, but numbers are treated as a unit
\`sin^-1(x)\`	`sin^-1(x)`	function names are treated as constants
\`d/dxf(x)=lim_(h->0)(f(x+h)-f(x))/h\`	`d/dxf(x)=lim_(h->0)(f(x+h)-f(x))/h`	complex subscripts are bracketed, displayed under lim
\`f(x)=sum_(n=0)^oo(f^((n))(a))/(n!)(x-a)^n\`	`f(x)=sum_(n=0)^oo(f^((n))(a))/(n!)(x-a)^n`	f^((n))(a) must be bracketed, else the numerator is only `a`
\`(x^2-2x-8)/(x-4)=x+2\`	`(x^2-2x-8)/(x-4)=x+2`	put parenthesis or brackets around entire numerator/demoninator
\`int_0^1f(x)dx\`	`int_0^1f(x)dx`	subscripts must come before superscripts
\`[[a,b],[c,d]]((n),(k))\`	`[[a,b],[c,d]]((n),(k))`	matrices and column vectors are simple to type
\`x/x={(1, if x!=0),(text(undefined), if x=0):}\`	$x/x={(1,if x!=0),(text(undefined),if x=0):}$	piecewise defined function are based on matrix notation
\`a//b\`	`a//b`	use // for inline fractions
\`(a/b)/(c/d)\`	`(a/b)/(c/d)`	with brackets, multiple fraction work as expected
\`a/b/c/d\`	`a/b/c/d`	without brackets the parser chooses this particular expression
\`((a*b))/c\`	`((a*b))/c`	only one level of brackets is removed; * gives standard product
\`sqrtsqrtroot3x\`	`sqrtsqrtroot3x`	spaces are optional, only serve to split strings that should not match
\`(:a,b:) and x lt y lt 1\`	`(:a,b:) and x lt ylt1`	the < character is problematic in XML, use 'lt' or put formula in a comment
\`(a,b]=\`	`(a,b]=`	grouping brackets don't have to match
\`abc-123.45^-1.1\`	`abc-123.45^-1.1`	non-tokens are split into single characters, but decimal numbers are parsed with possible sign
\`hat(ab) bar(xy) ulA vec v dotx ddot y\`	`hat(ab) bar(xy) ulA vec v dotx ddot y`	accents can be used on any expression (work well in IE)
\`bb.bbb(AB].cc(AB).fr.tt[AB].sf(AB)\`	`bb.bbb(AB].cc(AB).fr.tt[AB].sf(AB)`	font commands; can use any brackets around argument
\`stackrel"def"= or \stackrel{\Delta}{=}" "("or ":=)\`	`stackrel"def"= or \stackrel{\Delta}{=}" "("or ":=)`	symbols can be stacked
\`{::}_(\ 92)^238U\`	`{::}_(\ 92)^238U`	prescripts simulated by subsuperscripts