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- https://math.libretexts.org/Courses/Coastline_College/Math_C280%3A_Calculus_III_(Tran)/03%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/3.09%3A_Optimization_of_Functions_of_Several_VariablesThe application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier...The application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier sections of this chapter, the introduction of more independent variables leads to more possible outcomes for the calculations. The main ideas of finding critical points and using derivative tests are still valid, but new wrinkles appear when assessing the results.
- https://math.libretexts.org/Courses/Kansas_State_University/Your_Guide_to_Intermediate_Algebra/04%3A_Quadratic_and_Polynomial_Functions/4.05%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWe have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific vari...We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. To use the Quadratic Formula, we substitute the values of \(a,b\), and \(c\) from the standard form into the expression on the right side of the formula. The solutions to a quadratic equation of the form \(a x^{2}+b x+c=0, a \neq 0\) are given by the formula:
- https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/09%3A_Quadratic_Equations_and_Functions/904%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Highline_College/Math_084__Intermediate_Algebra_Foundations_for_Soc_Sci_Lib_Arts_and_GenEd/04%3A_Functions/4.07%3A_Quadratic_FunctionsYou may recall studying quadratic equations in Intermediate Algebra. In this section, we review those equations in the context of our next family of functions: the quadratic functions.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/14%3A_Differentiation_of_Functions_of_Several_Variables/14.08%3A_Maxima_Minima_ProblemsThe application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier...The application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier sections of this chapter, the introduction of more independent variables leads to more possible outcomes for the calculations. The main ideas of finding critical points and using derivative tests are still valid, but new wrinkles appear when assessing the results.
- https://math.libretexts.org/Workbench/MAT_2420_Calculus_II/07%3A_Parametric_Equations_and_Polar_Coordinates/7.06%3A_Conic_SectionsConic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection...Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle.
- https://math.libretexts.org/Courses/Mission_College/Math_3B%3A_Calculus_2_(Sklar)/11%3A_Parametric_Equations_and_Polar_Coordinates/11.05%3A_Conic_SectionsConic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection...Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle.
- https://math.libretexts.org/Workbench/Intermediate_Algebra_2e_(OpenStax)/09%3A_Quadratic_Equations_and_Functions/9.04%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/18%3A_Quadratic_Equations_and_Functions/18.04%3A_Solve_Quadratic_Equations_Using_the_Quadratic_FormulaWhen we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier w...When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. Mathematicians look for patterns when they do things over and over in order to make their work easier. In this section we will derive and use a formula to find the solution of a quadratic equation.
- https://math.libretexts.org/Courses/Montana_State_University/M273%3A_Multivariable_Calculus/14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/Optimization_of_Functions_of_Several_VariablesThe application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier...The application derivatives of a function of one variable is the determination of maximum and/or minimum values is also important for functions of two or more variables, but as we have seen in earlier sections of this chapter, the introduction of more independent variables leads to more possible outcomes for the calculations. The main ideas of finding critical points and using derivative tests are still valid, but new wrinkles appear when assessing the results.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_370%3A_Precalculus/02%3A_Linear_and_Quadratic_Functions/2.03%3A_Quadratic_FunctionsYou may recall studying quadratic equations in Intermediate Algebra. In this section, we review those equations in the context of our next family of functions: the quadratic functions.
