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The quadratic formula is exactly correct when performed using the idealized arithmetic of real numbers, but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available on computers, the limitations of the number representation ...
While a student at Harvard Business School, Bricklin co-developed VisiCalc in 1979, making it the first electronic spreadsheet readily available for home and office use. It ran on an Apple II computer, and was considered a fourth generation software program. VisiCalc is widely credited for fueling the rapid growth of the personal computer industry.
The form comes with two worksheets, one to calculate exemptions, and another to calculate the effects of other income (second job, spouse's job). The bottom number in each worksheet is used to fill out two if the lines in the main W4 form. The main form is filed with the employer, and the worksheets are discarded or held by the employee.
A backup of an Excel Spreadsheet Add-in (DLL) .xll: Adds custom functionality; written in C++/C, Fortran, etc. and compiled into a special dynamic-link library: Macro .xlm: A macro is created by the user or pre-installed with Excel. Template .xlt: A pre-formatted spreadsheet created by the user or by Microsoft Excel. Module .xlv
Al-Khwārizmī's method of solving linear and quadratic equations worked by first reducing the equation to one of six standard forms (where b and c are positive integers) squares equal roots (ax 2 = bx) squares equal number (ax 2 = c) roots equal number (bx = c) squares and roots equal number (ax 2 + bx = c) squares and number equal roots (ax 2 ...
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
That is, h is the x-coordinate of the axis of symmetry (i.e. the axis of symmetry has equation x = h), and k is the minimum value (or maximum value, if a < 0) of the quadratic function. One way to see this is to note that the graph of the function f ( x ) = x 2 is a parabola whose vertex is at the origin (0, 0).