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In the figure, Excel is used to find the smallest root of the quadratic equation x 2 + bx + c = 0 for c = 4 and c = 4 × 10 5. The difference between direct evaluation using the quadratic formula and the approximation described above for widely spaced roots is plotted vs. b.
[2] [3] The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of −1. [4] The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
[1] [2] [3] [better source needed]. For example, 3 x 2 − 2 x y + c {\displaystyle 3x^{2}-2xy+c} is an algebraic expression. Since taking the square root is the same as raising to the power 1 / 2 , the following is also an algebraic expression:
Equation Maker for Mac: Yes No No Yes No No No Mac Available on Mac App Store PDF, PNG No Equation Notepad for Android: Yes Yes No No No Yes Yes No For the Android mobile operating system. Can export LaTeX. EqualX Equation Editor: Yes Yes No Yes No Yes Yes No Windows, Linux Easy to use for beginners and advanced users. Highly customisable ...
By the fundamental theorem of algebra, if the monic polynomial equation x 2 + bx + c = 0 has complex coefficients, it must have two (not necessarily distinct) complex roots. Unfortunately, the discriminant b 2 − 4c is not as useful in this situation, because it may be a complex number. Still, a modified version of the general theorem can be ...
1988 Excel 1.5; 1989 Excel 2.2; 1990 Excel 3.0; 1992 Excel 4.0; 1993 Excel 5.0 (part of Office 4.x—Final Motorola 680x0 version [122] and first PowerPC version) 1998 Excel 8.0 (part of Office 98) 2000 Excel 9.0 (part of Office 2001) 2001 Excel 10.0 (part of Office v. X) 2004 Excel 11.0 (part of Office 2004) 2008 Excel 12.0 (part of Office 2008)
The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem.