There are a number of operations that one might be asked to do with quadratic equations, so it would be helpful if you could be more specific, but I'll take a stab.

I'll assume you are talking about an equation of the form:

y = c + b*x + a*x^2, where x, y, a, b, and c are real numbers, and x is the independent variable.

There are a couple of things to notice about this equation. First, because x^2 grows much faster than x when the magnitude of x is large, the curve that this equation describes grows toward either positive infinity or negative infinity, depending on the sign of a, as x goes to positive and negative infinity. That fact may help you to remember that the curve is a parabola (not a proof, mind you). Another helpful thing to notice is that when x = 0, y = c, and because x^2 is very small compared to x near x=0, you can approximate the equation as a line near x=0 (with slope b). With these two pieces of information, you can sketch the curve very roughly. That should help understand what to expect from further analysis (not perfectly, but it'll steer you away from really bad mistakes).

Now, what kinds of things does one do with a quadratic? One thing you can do is look for roots, i.e., values of x for which y = 0. That's usually what we mean when we say "Solve the quadratic." It turns out that quadratics can be solved explicitly with a little work. I'm not going to derive it, but there is a formula for the solution called, unoriginally, the Quadratic Formula. It's probably there in your book, and you can just plug in the coefficients a, b, and c to find the solutions. I'm not going to try to reproduce the mathematical notation in plain text. One caveat: there may not be any real roots (values of x for which y = 0). Think about the shape of a parabola and where you can place it on a graph relative to the x-axis, and you should see that there may be 2 places where the curve crosses the x-axis, 1 place where it just touches the x-axis (called a "double root"), or it may not touch or cross the x-axis at all. In the last case, there will be 2 complex roots, but that's beyond the scope of this discussion.

You may also be asked to factor the equation. That just means to reduce it to the product of terms of the form "g + h*x)", where g and h are real numbers. This amounts to finding the roots by intelligent guessing, rather than using the formula. It's essentially the inverse of multiplying two linear terms to get a quadratic, like:

y = (1 - 2*x)*(3 + 4*x) = 1*3 - 3*2*x + 1*4*x - 2*4*x*x = 3 - 2*x - 8*x^2

Factoring by inspection can work very well for quadratics with relatively small integer coefficients, but basically requires a lot of practice to get very good at.

Hope this all helps a little bit. Feel free to comment with more specific questions, and I'll try to help. There's more detail in the linked Wikipedia article.
Source(s):
http://en.wikipedia.org/wiki/Quadratic_equation


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