All quadratic functions with real coefficients can be graphed on a standard x-y graph.
Not all quadratic functions have real roots, maybe that's what you were thinking of?
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There are some relationships but not all relationships are always true. Any function can be represented by an equation. But all equations are not functions. For example, y = sqrt(x) is the equation of the square root relationship which can be graphed as a parabola on its side, but it is not a function. It has slopes at each point. Some functions can be plotted as graphs but not all. A function such as f(x) = 1 when x is rational, and f(x) = 0 when x is irrational has no slope and cannot be plotted as a graph. A graph of a vertical line is not a function.
No. Some have two solutions where as some have none.
No. Well, it depends what you mean with "any quadratic equation". The quadratic formula can solve any equation that can be converted to the form: ax2 + bx + c = 0 Note that it involves only a single variable. There are other limitations as well; for example, no additional operations. If a variable, or the square of a variable, appears in the denominator (1/x, or 1/x2), then some might say that it is "quadratic", but it might no longer be possible to convert the equation into the standard form named above. Similarly, if you have additional operations such as square roots or higher roots, trigonometric functions, etc., it might not be possible to convert the equation into a form that can be solved by the quadratic formula.
As you probably suspect, there are no non-mathematical situations in which you would use the quadratic formula.
Here are some methods you can use:* Trial and error. This works especially well if the solution is a small integer. * Factoring. You must first write the equation in such a form that you have zero on the right. * Completing the square. * Using the quadratic formula. The last two methods work in all cases. The quadratic formula is easier to work with in the general case.